CC BY
0Probl. Anal. Issues Anal. Vol. 13 (31), No 2, 2024, pp. 3-24
DOI: 10.15393/j3.art.2024.15810
3
UDC 517.547
N. Bharti
A NORMAL CRITERION CONCERNING SEQUENCE OF FUNCTIONS AND THEIR DIFFERENTIAL POLYNOMIALS
Abstract. In this paper, we study normality of a sequence of mero-morphic functions whose differential polynomials satisfy a certain condition. We also give examples to show that the result is sharp.
Key words: normal families, differential polynomials, meromor-phic functions
2020 Mathematical Subject Classification: 30D30, 30D35, 30D45, 34M05
1. Introduction. In what follows, H(D) and M.(D) are the classes of all holomorphic and meromorphic functions in the domain D c C, respectively. A family TcM.(D) is said to be normal in D if every sequence of functions in T has a subsequence that converges locally uniformly in D with respect to the spherical metric to a limit function, which is either meromorphic in D or the constant 8. In the case T c H(D), the Euclidean metric can be substituted for the spherical metric (see [25], [34]). The idea of the normal family is attributed to Paul Montel [22], [23]. Ever since its creation, the theory of normal families has been a cornerstone of complex analysis with far-reaching applications in dynamics of rational as well as transcendental maps, function theory of one and several variables, bicomplex analysis, harmonic mappings, complex projective geometry, functional analysis etc. (see [1], [2], [5], [6], [12], [15], [20], [31], [34]).
The main purpose of this paper is to study the normality of a sequence of non-vanishing meromorphic functions in a domain D c C, whose differential polynomials have non-exceptional holomorphic functions in D. For f,g e M.(D), if f (z) — g(z) ^ 0 in D, then g is said to be an exceptional function of f in D. On the other hand, if there exist at least one z e D for which f (z) — g(z) = 0, then g is said to be a non-exceptional function
© Petrozavodsk State University, 2024
of f in D. If g happens to be a constant, say k, then k is said to be an exceptional (respectively, non-exceptional) value of f in D.
Definition 1. [16] Let k e N, f e M(D) and n0,nl,... ,nk be nonnegative integers, not all zeros. By a differential monomial of f we mean an expression of the form
M[f] := a •(/)no (fT1 (f'T •••(/(k)Tk,
where a (1 0, 8) e M.(D). If a is taken to be the constant function 1, then we say that the differential monomial M[/] is normalized. Further, the quantities
k k
\m :=yj ni and ym := + 1)%
j"0 j"0
are called the degree and weight of the differential monomial M[f], respectively.
k n-
For 1 ^ i ^ m, let Mi[f] = (/pj)) be m differential monomials
j "0
of f. Then the sum
m
P[f] := ^ *iMi[f ]
i"l
is called a differential polynomial of f and the quantities
Xp := max {XMi: 1 ^ i ^ m} and := max {^Mi: 1 ^ i ^ m}
are called the degree and weight of the differential polynomial P[f], respectively. If XMl = XM2 = • • • = XMm, then P [f ] is said to be a homogeneous differential polynomial.
In this work, we are concerned with the homogeneous differential polynomials of the form
Q[f] := fx°(fxi)Pyiq(fx2)py2) • • • (fXk)q, (1)
where x0, xl, ..., xk, yl, y2, ..., yk are non-negative integers, such that xi ^ yi for i = 1, 2, ..., k.
The differential polynomial (1) first appeared in the literature in [14] and has been used extensively since then, particularly in finding normality criteria of families of meromorphic functions (see [26], [27], [28]).
k k
We set x' = XI xi and y' = XI Vi- Further, we assume that x0 > 0 and
y' > 0- Using the generalized Leibniz rule for derivatives, one can easily verify that
(fxi\(Vi) = V1 y*'_ f (nl) j (n2q ... j pnXi),
^ n1!n2! ■■■ nx.! '
where ni,s are non-negative integers. Thus, the degree of Q[f], Aq = x0 + x' and the weight of Q[f], ¡jq = x0 + x' + y' = Xq + y'-
2. Motivation and main results. In [19, Problem 5-11], Hayman posed the following problem:
Problem A. Let T c M.{D) and k be a positive integer. Suppose that for each f e T, f (z) ^ 0, f(k) ^ 1. Then, what can be said about the normality of T in D?
Gu [17] considered Problem A and confirmed that the family T is indeed normal in D. Subsequently, Yang [29] proved that the exceptional value 1 of f(k) can be replaced by an exceptional holomorphic function-Chang [3] considered the case when f(k) — 1 has limited number of zeros and obtained the normality of T- Thin and Oanh [28] replaced f(k) with a differential polynomial of f - Later, Deng et al- [11] established that there is no loss of normality even when f(k) — h has zeros for some h e 'H(D) as long as the number of zeros are bounded by the constant k- Chen et al- [8] took a sequence of exceptional holomorphic functions instead of a single exceptional holomorphic function- Recently, Deng et al- [13] proved the following theorem concerning a sequence of meromorphic functions:
Theorem B. Let {fj} c M.(D) and {hj} c H(D) be sequences of functions in D. Assume that h j ^ h locally uniformly in D, where h e H(D) and h 1 0. Let k be a positive integer. If, for each j, fj(z) ^ 0 and f(k — h j(z) has at most k distinct zeros, ignoring multiplicities, in D, then {fj} is normal in D.
Following Thin and Oanh [28], a natural question about Theorem B arises:
Question C. Let {fj} c M.(D) and {hj} c H(D) be sequences of functions in D. Is it possible to generalize Theorem B for differential polynomials Q[fj ]?
In this paper, our first objective is to find a complete answer to Question C. Since normality is a local property, one can always restrict the domain to the open unit disk D.
Theorem 1. Let {fj} c ^(D) and {h j} c H(D) be such that h j ^ h locally uniformly in D, where h e H(D) and h 1 0. Let Q[fj] be a differential polynomial of fj as defined in (1), having weight /jq. If, for each j, fj(z) ^ 0 and Q[fj] — h j has at most ¡jq — 1 zeros, ignoring multiplicities, in D, then {fj} is normal in D.
Remark 1. Theorem 1 gives an affirmative answer to Question C.
Our next objective is to find whether the upper bound for the number of zeros of Q[fj] — hj in Theorem 1 can be improved. In view of this, we obtain the following result, which is more general than Theorem 1:
Theorem 2. Let {fj} c ^(D) be a sequence, such that, for each j, fj has poles of multiplicity at least m, m e N. Let {h j} c H(D) be such that h j ^ h locally uniformly in D, where h e H(D) and h 1 0. Let Q[fj] be a differential polynomial of fj as defined in (1), having degree Xq and weight /jq. If, for each j, fj(z) ^ 0 and Q[fj] — h j has at most I^q + Xq(m — 1) — 1 zeros, ignoring multiplicities, in D, then {fj} is normal in D.
Remark 2. Clearly, if we do not take the multiplicity of poles of fj into account, then Theorem 2 reduces to Theorem 1.
As a direct consequence of Theorems 1 and 2, we have
Corollary 1. Let {fj} c ^(D) and {hj} c H(D) be such that hj ^ h locally uniformly in D, where h e H(D) and h 1 0. If, for each j, fj(z) ^ 0 and Q[fj](z) ^ hj(z), then {fj} is normal in D.
In the following, we show that the condition 'fj(z) ^ 0' in Theorem 2 is essential.
Example 1. Consider a sequence {fj} c ^(D) given by fj (z) = jz, j e N, j ^ 2. Let Q[fj ] := fj fj, so that ¡jq = 3, and let hj(z) = z. Then hj ^ z 1 0 and Q[fj](z) — h j(z) has at most one zero in D. However, {fj} is not normal in D.
Taking h j(z) = 1/z in Example 1, we find that h j cannot be meromor-phic in D. Furthermore, the condition "h 1 0" in Theorem 2 cannot be dropped as demonstrated by the following example:
Example 2- Let {fj} c ^(D) be such that fj(z) = ejz, j e N, and let hj " 0, so that hj — h " 0- Let Q[fj] be any differential polynomial of fj of the form (1)- Clearly, Q[fj](z) — h(z) has no zero in D- But the sequence {fj} is not normal in D-
The following example establishes the sharpness of the condition "Q[fj] — h j has at most ¡jq + \Q(m — 1) — 1 distinct zeros in D" in Theorem 2:
Example 3- Let {fj} c ^(D) be such that
fj(z) = > 3,j e N,
jz
and let Q[fj ] : = fj fj - Then xq = 2, ¡jq = 3, m = 1, and Q[fj](z) = —1/j2 z3- Consider hj(z) = 1/(z — 1)3, so that {hj} e H(D) and h j — 1/(z — 1)3 1 0- Then, by simple calculations, one can easily see that Q[fj](z) — h j(z) has exactly ¡jq + XQ(m — 1) = 3 distinct zeros in D-However, the sequence {fj} is not normal in D-
3. Preliminary results. What follow are the preparations for the proof of the main result- We assume that the reader is familiar with standard definitions and notations of Nevanlinna's value distribution theory, like m(r,f ),N(r, f ),T(r, f),S(r,f) (see [18], [30])- Recall that a function g e M(C) is said to be a small function of f e M.(C) if T(r,g) = S(r, f) as r possibly outside a set of finite Lebesgue measure-
Notation: By Dr(a), we mean an open disk in C with center a and radius r- D = ^1(0) is the open unit disk in C-
The following lemma is an extension of the Zalcman-Pang Lemma due to Chen and Gu [9] (cf- [24, Lemma 2])-
Lemma 1- (Zalcman-Pang Lemma) Let T c ^(D) be such that each f e T has zeros of multiplicity at least m and poles of multiplicity at least p. Let —p < a < m. If T is not normal at z0 e D, then there exist sequences {fj} c T, {zj} c D, satisfying Zj ^ z0, and positive numbers Pj with pj ^ 0, such that the sequence {gj} defined by
9j«) = PJafj (zj + Pj C)- g(()
locally uniformly in C with respect to the spherical metric, where g is a non-constant meromorphic function on C, such that for every ( e C, 9#(C) ^ 9#(0) = 1.
We remark that if f(z) ^ 0 in D for every f e T, then a e (—p, + 8). Likewise, if each f e T does not have any pole in D, then a e (—8, m), and if f(z) ^ 0, 8 in D for every f e T, then a e (—8, + 8).
Lemma 2. [10, Lemma 3] Let T c ^(D) and suppose that h e H(D) or h "8. Further, assume that for each f e T, f(z) ^ h(z) in D. If T is normal in D\ {0} but not normal in D, then there exists a sequence {fj} c T, such that fj ^ h in D\ {0}.
Proposition 1. Let f e M.(C) be a transcendental function and let Q[f] be a differential polynomial of f as defined in (1), having degree Xq and weight /jq. Assume that ^ (1 0, 8) is a small function of f. Then
XqT(r, f) ^ N(r, f) + (1 + — Xq)n(t, -)n(t, ) + 5(r, f).
Proof. By definition of Q[f], it is apparent that Q[f] 1 0. Then, from the first fundamental theorem of Nevanlinna, we have
7) +xq>n (r7-
el/I) + XqN (r't
From Nevanlinna's theorem on logarithmic derivative, we find that
XqT(T, f) = Xq m(r, y) + XqN(t, ^ + 0(1) ^
^ m( ^ )+ m(r, om)+ xqn (^ 7)+ 0(1).
m\ r, ) = S(r, f).
Thus, from (2), we obtain
XqT(r, f) ^ m(r, ^-J]) + XqN(t, i) + S(r, f) =
= T(r, Q[f]) — N(r, ^j]) + XqN(r, 1) + S(r, f).
Applying the second fundamental theorem of Nevanlinna for small functions to T(r,Q[f]), we get
r\ ^ + N(r,Q[f]) + + N{r
XqT(r, f) ^ XqN(t, -) + N(r,Q[f]) + N(r, ) +n(j
—
fj y}Q[f]J V ' Q[f] —
1
QUI
<r,W\ )+S (f)
xqn(t,^ +Npr, f) + N(r,-1-)+N(r 1
—
f> x ' ^'QUV v>Q[f
(3)
—N (r ,m)(f,f)-
Since a zero of f with multiplicity m is also a zero of Q[f] with multiplicity at least (m + 1)Aq — ¡jq ,
Therefore, from (3), we obtain XqT (r,f) ^ \qN(t-, 1) + N (r, f) + [1 + — (m + 1) Aq] n(t, 1) +
+¥ (r ,wh)(r,;) ^
^ N(r, f) + (1 +Hq — Xq)n{t, 1) + N{r, 1 + S(r, f).
□
Corollary 2- Let f e M.(C) be a transcendental function and let Q[f] be a differential polynomial of f as defined in (1). Assume that ^ (1 0, 8) is a small function of f. If f ^ 0, then Q[f] — ^ has infinitely many zeros in C.
Proof. From Proposition 1, we have
W 1 ) + N ( r,-1
_ (4)
Since f ^ 0, N(r, 1/f) = 0. Thus, from (4), we obtain
\QT p r, f) ^ N p r, f) + pi +ßQ - Xq)n(t, -) +N(r, ) + 5 p r, f).
XqTpr, f) ^ Npr, f) + N(r, ) + 5pr, f).
This implies that
1
Q[F]-^
pXq - 1)Tpr, F) ^ N(r, ) + 5pr, F).
Since Xq— 1 > 0, it follows that Q[F]—vp has infinitely many zeros in C. □
In [3], Chang proved that if f is a non-constant rational function, such that f ^ 0, then for k ^ 1, fpfcq — 1 has at least k + 1 distinct zeros in C. Using the method of Chang [3], Deng et al. [11] proved that the constant 1 can be replaced by a polynomial p (1 0). Recently, Xie and Deng [32] sharpened the lower bound for the distinct zeros of fpfcq —p in C by involving the multiplicity of poles of /. Thin and Oanh [28] extended the result of Chang to differential polynomials by proving that if f (^ 0) is a non-constant rational function, then Q[f] — 1 has at least Jq distinct zeros in C. We obtain a better result in the following form:
Proposition 2. Let f be a non-constant rational function, having poles of multiplicity at least m, m e N, and let p (1 0) be a polynomial. Let Q[f] be a differential polynomial of f as defined in (1), having degree Xq and weight jq. Assume that f ^ 0. Then Q[f] — p has at least Jq + XQ(m — 1) distinct zeros in C.
Proof. Since f ^ 0, it follows that f cannot be a polynomial and, so, f has at least one pole. Therefore, we can write
m = -n—C-. (5)
Y\(z + ai)ni+m-1
Let
p(z) = C2 ^(z + Pi)k, (6)
i"l
where Cl} C2 are non-zero constants; I, n, ni are positive integers; and U are non-negative integers. Also, fa (when 1 ^ i ^ /) are distinct complex numbers and o^ (when 1 ^ i ^ n) are distinct complex numbers. From (5), one can deduce that
QLf](z) " s-^-, (7)
Z + a )xQpni +m-l)+^Q-xQ
i-1
where hq is a polynomial of degree (n — 1)(hq — Xq).
Also, it is easy to see that Q[f] —p has at least one zero in C. Therefore, we can set
Cs ft (z + *
Q[f](z) = p(z) + i=1 ^
YYi"l(z + ai)xQpni+m-iq+^Q-xQ
where C3 e C\ {0}, qi are positive integers, and ^ (1 ^ i ^ q) are distinct complex numbers-
l n
Let L = X h and N = X ni- Then from (6), (7) and (8), we have i"1 i"1
n
C2 ^(z + Pi)k n(z + ^)XQ(n*+m-1)+»Q-XQ +C3 + 7i)9i = hQ(z). (9) i"1 i" 1 i" 1
From (9), we find that
Y^^i " Zj [XQ(ni + m — 1)+ VQ — ^Q] + 2 h = "1 "1 "1
= xqn + n(m — 1) Aq + n(/iQ — \q) + L
and C3 = —C2-
Also, from (9), we get
n
n(1+Pir )li n(1 +a%r )xi(ni+m-1)+^-xi — ^(1+-jiV)qi = "1 "1 "1 = rHQ + ^Q( N+n(m-1)-1)+L ¡)(r)^
where b(r) := r(n-1)(^Q-XQ) hQ(1/r)/C2 is a polynomial of degree at most (n — 1)(^q — xq)- Furthermore, it follows that
npi + ßiT ) kU pi + )Xq (ni+m-l)+^Q-XQ __
npi +liT ) * i"l
" 1 + ,
rHQ + \o(N+n(m-1)-1) + L h(r\
L_°SL± = 1 + 0 (r^Q+XQ(N+n(m-1)-1) + LJ (10)
n(1 + W)qi "1
as r — 0- Taking logarithmic derivatives of both sides of (10), we obtain kPi [XQ(ni + m — 1) + Vq — Xq}^ yi qi^i
Sp L ißi Sp IXQpni -r m — i) r pq - xq j Ui yi ~l i - ßir i + ®ir ~l i +lir
" o pN+n(m-l)-l)+L-1^ as r ^ 0. (11)
Let Si = {[i,[2,...,[} x {ai,a2,... ,an} and S2 = {[i,[2,..., [l} x {li,12,..., lq}. Consider the following cases: Case 1: Si = S2 = 0. Let an+i = [i when 1 ^ i ^ I and
N _ \\Q(ni + m — 1) + Iq — Xq, if 1 ^i^n, i ) Ii_n, ifn + 1 ^ i ^ n + 1.
Then (11) can be written as
n+l q
y ^iai —y qw = 0 , ^Q+xQ(N+n(m-i)-i)+L-u as r^ 0 (12) 1 + air " 1 + v '
Comparing the coefficients of rj, j = 0,1,..., iq + Xq(N + n(m — 1) — 1) + L — 2 in (12), we find that
n+
Y^Niai—2 qai = 0, for each j = 1, 2,..., iq+Xq(N+n(m—1) — 1)+L—1. i"i i"i
(13)
Now, let an+l+i = ^ for 1 ^ i ^ q. Then, from (13) and the fact that
n+l q
ZN — qi = 0, we deduce that the system of equations
i"i i"i
n+ +
2 aXi = 0,j = 0, 1,...,Iq + Xq(N + n(m — 1) — 1) + L — 1, (14) i"i
has a non-zero solution
(xi , . . . , Xn+l, Xn+l + i , . . . , Xn+l+qq = (N1, . . . ,Nn+l, — Ql, . . . , — qqq .
This is possible only when the rank of the coefficient matrix of the system (14) is strictly less than n + I + q.
n
Hence, iq + Xq(N + n(m — 1) — 1) + L < n + 1 + q. Since N = ^ini ^ n
i"i
l
and L = h ^ I, it follows that q ^ iq + XQ(m — 1).
i"i
Case 2: Si ^ 0 and S2 = 0.
We may assume, without loss of generality, that Si = {[i, [2,..., [S1}. Then [i = ai for 1 ^ i ^ si. Take s3 = I — si.
Subcase 2.1: s3 ^ 1.
Let an+i = Psi+i for 1 ^ i ^ s3. If ^ < n, then let
XQ(ni + m - 1) + ßQ -Aq + U, if 1 ^ i ^ si,
Ni " ' AQ(n + m - 1) + ßQ - Aq, if si + 1 ^ i ^ n,
J Sl-n+i, if n+ 1 +
If ^ = n, then we take
if n + 1 ^ i ^ n + s3.
AQpni + m - 1) + ßQ - Aq + U, if 1 ^ i ^ si,
Is-i-n+i, if n + 1 ^ i ^
if n + 1 ^ i ^ n + s3.
Subcase 2.2: s3 = 0.
If ^ < n, then set
AQpni + m - 1) + ßQ - Aq + U, if 1 ^ i ^ si, AQpni + m - 1) + ßQ - Aq, if si + 1 ^n
and if ^ = n, then set N = Aq{;ni+m—1)+ ¡q—Aq+U, for 1 ^ i ^ ^ = n. Thus, (11) can be written as:
where 0 ^ s3 ^ I — 1. Proceeding in the similar fashion as in Case 1, we deduce that q ^ ¡¡q + m — 1. Case 3: S1 = 0 and S2 ^ 0.
We may assume, without loss of generality, that S2 = {/31, 32,..., 3S2}. Then 3i = li for 1 ^ i ^ s2. Take s4 = I — s2. Subcase 3.1: s4 ^ 1.
Let ig+i = 3S2+i for 1 ^ i ^ s4. If s2 < q, then set
Qi
qi - li, if 1 ^ i ^ s 2, qi, if s 2 + 1 ^ i ^ q,
-I s2-q+i, if q + 1 ^q + S4.
If s2 = q, then set
if 1 ^ i ^ s 2,
if q + 1 ^ i ^ q + s4.
Subcase 3.2: s4 = 0.
If s2 < q, then set
Qi =
i
k if 1 ^ i ^ s2, if s2 + 1 ^ % ^ q,
and if s2 = q, then set Qi, = qi — U, for 1 ^ i ^ s2 = q. Thus, (11) can be written as:
£
i"i
[Xq(v4 + m — 1) + Iq — Xq] ai
q+S4 2
Qi i
1 + a^ 1 + tir
= 0 (r+n(™-i)-i)+L-i) as r ^ 0,
where 0 ^ s4 ^ I — 1. Proceeding in the similar way as in Case 1, we deduce that q ^ Iq + Xq (m — 1). Case 4. Si ^ 0 and S2 ^ 0.
We may assume, without loss of generality, that Si = {[i, [2,..., [S1} , S2 = {7i,T2,... ,1s2}. Then [i = ai for 1 ^ i ^ si and 7i = [s1+i for
1 ^ i ^ s2. Take s5 = I — s2 — si. Subcase 4.1: s5 ^ 1.
Let an+j, = uSl+S2+i for 1 ^ i ^ s5 and if si < n, then set
Ni = ^
XQ(ni + m — 1) + Iq — Xq + k Xq (ni + m — 1) + Iq — Xq ,
S1 + S2 -n+i,
If si = n, then set
Ni =
XQ(ni + m — 1) + Iq — Xq + 1.
1 + 2 - n+ ,
if 1 ^ i ^ si, if si + 1 ^ i ^ n, if n + 1 ^ i ^ n + s5.
if 1 ^ i ^ si,
if n + 1 ^ i ^ n + s5.
If 2 < , then set
Qi =
{qi I s1+i, i,
if 1 ^ i ^ s2, if s2 + 1 ^ % ^ q,
and if s2 = q, then set Qi = qi — lS1+j,, for 1 ^ i ^ s2. Subcase 4.2: s5 = 0.
If i < n, then set
Ni =
XQ(ni + m — 1) + Iq — Xq + k, if 1 ^ i ^ si
XQ(ni + m — 1) + Iq — X
if si + 1 ^ i ^ n.
If Si = n, then set Ni = XQ(ni + m — 1) + Jq — Xq + h for 1 ^ i ^ si. Also, if s 2 < q, then set
i:
q _ , Qi - h1+i, if 1 ^ i ^ S2, ' qi, if S 2 + 1 ^ i ^ Q,
and if s 2 = q, then set Qi = qi — lSl+i, for 1 ^ i ^ s2. Thus, in both subcases, (11) can be written as
n+ «5 Ar q n
y l^qj, —y Qi^i " 0 + pN+n(m-1)-1) + L-1\ as r ^ 0
ir! i + qjr ~11 + iir '
where 0 ^ s5 ^ I — 2. Proceeding in the similar fashion as in Case 1, we deduce that q^ jq + XQ(m — 1). □
Lemma 3. Let {fj} c ^(D) be a sequence of non-vanishing functions, all of whose poles have multiplicities at least m, m e N. Let {hj} c H(D) be such that hj ^ h locally uniformly in D, where h e H(D) and h(z) ^ 0 in D. If, for each j, Q[fj] — hj has at most jq + XQ(m — 1) — 1 zeros, ignoring multiplicities, in D, then { j} is normal in D.
Proof. Without loss generality, suppose that {fj} is not normal at 0 e D. Then, by Lemma 1, there exists a sequence of points {Zj} c D with Zj ^ 0, a sequence of positive real numbers satisfying pj ^ 0, and a subsequence of {fj}, again denoted by { fj}, such that the sequence
F (t\ — ^(zj ^ Pj^
Fj(hq-\q)/\Q ^ F j
spherically locally uniformly in C, where F e M.(C) is a non-constant and non-vanishing function having poles of multiplicity at least m. Clearly, Q[Fj] ^ Q[F] spherically uniformly in every compact subset of C disjoint from poles of F. Also, one can easily see that Q[Fj](() = Q[fj](Zj + pj(). Thus, for every ( e C\{F-1(œ)},
Q[ fj ]( ^ +P3( ) — hj( z3 + pj ( ) = Q[Fj ](0 — h3( z3 + pj( )^Q[F ](0 — h(0)
spherically locally uniformly. Since F is non-constant and x0 > 0, Xi ^ yi for all i = 1, 2, ..., k, by a result of Grahl [16, Theorem 7], it follows that Q[F] is non-constant. Next, we claim that Q[F] — h(0) has at most Jq + XQ(m — 1) — 1 zeros in C.
Suppose, on the contrary, that Q[F] — h(0) has ¡q + AQ(m — 1) distinct zeros in C, say Q,i = 1, 2,..., ¡Q + AQ(m—1). Then by Hurwitz's theorem, there exit sequences , i = 1, 2, ..., ¡q + AQ(m — 1) with ^ Q, such that for sufficiently large j, Q[fj](Zj + Pj(j,i) — hj(Zj + Pj(j,i) = 0 for i = 1, 2, ..., ¡q + AQ(m — 1). However, Q[fj] — hj has at most ¡q + AQ(m — 1) — 1 distinct zeros in D. This proves the claim. Now, from Corollary 2, it follows that F must be a rational function which contradicts Proposition 2. □
Proposition 3. Let t be a positive integer. Let {fj} c ^(D) be a sequence of non-vanishing functions, all of whose poles have multiplicities at least m, m e N, and let {hj} c H(D) be such that hj ^ h locally uniformly in D, where h e H(D) and h(z) ^ 0. If, for every j, Q[fj](z) — zthj(z) has at most ¡q + AQ(m — 1) — 1 zeros in D, then {fj} is normal in D.
Proof. In view of Lemma 3, it suffices to prove that T is normal at z = 0. Since h(z) ^ 0 in D, it can be assumed that h(0) = 1. Now, suppose, on the contrary, that {fj} is not normal at z = 0. Then, by Lemma 1, there exists a subsequence of { j}, which, for simplicity, is again denoted by {fj}, a sequence of points {Zj} c D with Zj ^ 0, and a sequence of positive real numbers satisfying pj ^ 0, such that the sequence
F (t\ = ^(Zj + Pj^) F(t\
Fj(t+P,Q-\Q)/\Q ^ F j
spherically locally uniformly in C, where F e M.(C) is a non-constant function. Also, since each fj is non-vanishing, it follows that F is non-vanishing. We now distinguish two cases.
Case 1: Suppose that there exists a subsequence of Zj/pj, again denoted by Zj/pj, such that Zj/pj ^8 as j ^ 8. Define
3j(() •= fj(Zj + Zj(). Then an elementary computation shows that
Q[g3 ](0 = z-Q[ f3 ](z3 + z3c),
and, hence,
Q[ fj](Zj + ZjC) — (Zj + ZjC) %j( Zj + ZjC) =
= z)Q[g3](() — (Zj+ZjC)hj(Zj+Zj() = zj [Q[g3](<) —1 + C)4h3(z3 + Zj()] .
Since (1+0f'hj(Zj+Zj() ^ (1+()* ^ 0 in D, and Q[fj](Zj+Zj()-zj(1 + 0fhj(Zj + Zj0 has at most jq + \q(m — 1) — 1 zeros in D, by Lemma 3, it follows that {gj} is normal in D and, so, there exists a subsequence of {gj}, again denoted by {gj}, such that gj ^ g spherically locally uniformly in D, where g p ^(D) or g " 8. If ^ "8, then
f,(0 " p-1-'^-^ft(^ + ftc)
" (^y«-«-*./^(Z )"
Zj\ pt+^Q-XQq/xQ ~{t+hq-> ,P] ; ^ ^
/ Zi\(t+^Q-XQ)/XQ fPj A
converges spherically locally uniformly to 8 in C, showing that F "8: a contradiction to the fact that F is non-constant. Since gj(() ^ 0, it follows that either g(Q ^ 0 or g " 0. If g(() ^ 0, then, by the previous argument, we find that F " 8: a contradiction. If g " 0, then choose n e N, such that n + 1 > (t + jq — Xq)/(Xq). Thus, for each ( e C, we have
Fpn+1) (o = p-t+^Qy^^1 fjjn+1\ Zj + Pj( q =
= / Pl\-pt+^Q-XQq/xQ+n+1 g(n+i)\P±
Therefore, Fjn+1 (() ^ 0 spherically uniformly, which implies that F is a polynomial of degree at most n: a contradiction to the fact that F is non-constant and non-vanishing meromorphic function. Case 2: Suppose that there exists a subsequence of Zj/pj, again denoted by Zj/pj, such that Zj/pj ^ a as j ^ 8, where a e C. Then
Gj(0 " pjfj(Pj(q " Fj(( — ^ F(( — aq := G(0,
spherically locally uniformly in C. Clearly, G((q ^ 0. Also, it is easy to 0 = PJ'QI fj ](Pji
see that Q[G,](<) = pj'QU,](p,(). Thus,
Q[G,](0 - (%(p,() " Q[fj](pj0 ^ Q[G](0 - c
Pj
spherically uniformly in every compact subset of C disjoint from the poles of G. Clearly, Q[G](£) 1 (t, otherwise G has to be a polynomial, which is not possible since G(() ^ 0. Since Q[fj](pj() — (pj()fhj(pj() has at most Iq + Aq(m — 1) — 1 distinct zeros in D, it follows that Q[G](() — (m has at most iq + AQ(m — 1) — 1 distinct zeros in C and, hence, by Corollary 2, G must be a rational function. However, this contradicts Proposition 2. Hence T is normal in D. □
4. Proof of Theorem 2. By virtue of Lemma 3, it is sufficient to prove the normality of {fj} at points z p D, where h(z) = 0. Without loss of generality, assume that h(z) = zf'a(z), where t p N, a p V-(D), a(z) ^ 0 and a(0) = 1. Further, since hj — h locally uniformly in D, we can assume that
hj(z) = (z — zhl)tl(z — Zj,2)i2 --^z — Zj,i)tlaj(z),
i
where U = t, Zjti — 0 for 1 ^ i ^ I and aj (z) — a(z) locally uniformly
i-l
in D. Again, we may assume that Zj;1 = 0, since {fj(z)} is normal in D if and only if {fj(z + Zji)} is normal in D (see [25, p. 35]). Now, let us prove the normality of {fj} at z = 0 by applying the principle of mathematical induction on .
Note that if t = 1, then I = 1 and, so, hj(z) = zaj(z). Thus, by Proposition 3, {fj} is normal at z = 0. Also, if I = 1, then h j(z) = ztaj (z), and, again by Proposition 3, {fj} is normal at z = 0. So, let I ^ 2 and for n p N with 1 <t <n, suppose that {fj} is normal at z = 0. In accordance with the principle of mathematical induction, we only need to show that {fj} is normal at z = 0 when n = t.
Rearranging the zeros of hj, if necessary, we can assume that I zji I ^ I zj,i I for 2 ^ i ^ /. Let Zj,i = Wj. Then Wj — 0. Define
/ \ fi (wjz) , , hj(wjz) ^ , ,
9i(z) := JJ( j ' and Vj(z) := j( j ), z p Dr.(0), rj — 8. Wj
Then an easy computation shows that Q[gj](z) = wJ*Q[fj 1](wjz) and
Vj (z) " ztl (z — W)i2... (z — ) "" (z — 1)* aj (Wjz) - v(z)
locally uniformly in C. Clearly, 0 and 1 are two distinct zeros of and, hence, all zeros of v have multiplicities at most t — 1. Since
Qfe ](,)- * (,)-Q[ fj ](Wj ZW;hj {WjZ > (15)
wj
and Q[fj](wjz) — hjiwjz) has at most jq + \q{rn — 1) — 1 distinct zeros, it follows that Q[gj](z) — Vj(z) has at most jjq + \Q(m — 1) — 1 distinct zeros in C. Thus, by induction hypothesis, we find that {gj} is normal in C. Suppose that gj ^ g spherically locally uniformly in C. Then either g e M(C) or g "8. Case 1: ge M(C).
Since gj(z) ^ 0, it follows that either g(z) ^ 0 or g " 0. First, suppose that g(z) ^ 0. Since gj ^ g spherically locally uniformly in C, it follows that Q[gj] ^ Q[g] in every compact subset of C disjoint from the poles of g. Then, from (15), we find that Q[g] — v has at most jjq + \Q(m — 1) — 1 distinct zeros in C and, thus, by Corollary 2 and Proposition 2, g has to be a constant.
Next, we claim that {fj} is holomorphic in D$/2(0) for some 5 e (0,1). Suppose, on the contrary, that {fj} is not holomorphic in D$/2(0) for any 5 e (0,1). Then there exists a sequence rjj e D$/2(0), such that rjj ^ 0 and fj (rjj) = 8. Assume that rjj has the smallest modulus among the poles of fj. It is easy to see that rjj/wj ^ 8, otherwise
fj(rjj) = wjt+tiQ xq^/xqgj(rjj/wj) ^ 0, a contradiction.
Let
^ (z)and ^(z) 'h
, zeDrj (0), r, —> 8.
Then
and
QbPj ](z)-ui (z) = ^^js(lj
Q[ fj ]( Vj z) - hj( rjj z)
v)
:i6)
„ - f)-...(.
WAtl < \ t
-j) dj(rjjz) — z j
locally uniformly in C. From Lemma 3, it follows that {^j} is normal in C\ {0}. Since ipj (z) ^ 0 and ipj is holomorphic in D, one can easily see that {^j} is normal in D and, hence, in C. Assume that ^j ^ ^ spherically locally uniformly in C, where ^ e M(C) or ^ " 8. Since
(o) =__= (Wi)
j
Wj \pt+ßQ-XQ)/xQ
9j(0) — 0,
therefore, p 1 8. Also, since pj(z) ^ 0, we have p(z) ^ 0 or p " 0. However, the latter is not possible since 8 = pj(1) ^ p(1) = 8. Thus, p(z) ^ 0. Note that Q[pj](z) — Uj(z) ^ Q[p](z) — zt spherically uniformly in every compact subset of C disjoint from the poles of p, so, by (16), we conclude that Q[p](z) — z1 has at most jq + Xq(m — 1) — 1 distinct zeros in C. By Corollary 2 and Proposition 2, p reduces to a constant, which contradicts the fact that 8 = pj(1) ^ p(1) = 8. Hence, {fj} is holomorphic in D$/2(0). Since fj(z) ^ 0, it follows that {fj} is normal at z = 0.
Next, suppose that g " 0. Then, by the preceding discussion, one can easily see that {fj} is holomorphic in D$/2(0) and, hence, {fj} is normal at z = 0. Case 2: g " 8.
Let fa(z) := fj(z)/Zpt+^Q-XQ)/XQ. Then 1/0j(0) = 0. Subcase 2.1: When {1/pj} is normal at z = 0.
Then {pj} is normal at z = 0 and, so, there exists r > 0 with Dr(0) c D, such that {pj} is normal in Dr (0). Assume that pj ^ p spherically locally uniformly. Since pj(0) = 8, there exists p > 0, such that, for sufficiently large j, \pj(z)\ ^ 1 in Dp(0) c Dr(0). Also, since fj(z) ^ 0 in Dp(0), 1/fj is holomorphic in Dp(0) and, hence,
2 \Pt + HQ-\Q)/\Q _
1 1 1
fj (z) pj (z) Zpt+VQ-XQ)/XQ
[-) in bdp/2(0).
^p
Then, by the maximum principle and Montel's theorem [25, p.35], we
conclude that {fj} is normal at z = 0.
Subcase 2.2: When {1/pj} is not normal at z = 0.
By Montel's theorem, it follows that, for every e > 0, {1/ pj(z)} is not locally uniformly bounded in De(0). Therefore, we can find a sequence tj ^ 0, such that 1/ pj(tj) ^ 8. Since \1/ pj \ is continuous, there exists bj such that \1/pj (bj)\ = 1. Let
_fj(z) ^ ^ n _ „ ._ h j(biz)
t+
Then Kj (z) ^ 0 and a simple computation shows that
Kj(z) := u, —{ wN , z p Dr.(0), Tj ^ 8 and qj(z) := ,,
3K ' ^+P-Q-XQ)/XQ rjW> j H3 \ / ^t
0]M - and rll (Z) - sp — *...(, —1)% (V
Note that
(h.^ fj(bj) fj(bj) . (k\pt+t*Q-xQ)/xQ ^ œ
Since |1/ (j(bj)| = 1 and (t + jq — \q)/\q > 0, it follows that bj/wj ^ 8, and, hence, wj/bj ^ 0. This implies that qj(z) ^ zl locally uniformly in C. Further, since Q[Kj](z) — qj(z) = (Q[fj](bjz) — hj(bjz))/bj, it follows that Q[Kj] — qj has at most jq + \Q(m — 1) — 1 distinct zeros in C and, hence, by Lemma 3, {Kj} is normal in C\{0}. We claim that {Kj} is normal in C. Suppose otherwise. Then, by Lemma 2, there is a subsequence of {Kj}, which for the sake of convenience, is again denoted by {Kj}, such that Kj(z) ^ 0 in C\{0}, which is not possible since iKj(1)| = 1. This establishes the claim. Now, suppose that Kj ^ K spherically locally uniformly in C. It is evident that K(z) ^ 0 in C and K 1 8, as K(1) = 1. Then Q[Kj] ^ Q[K] spherically uniformly in every compact subset of C disjoint from the poles of K. Since Q[Kj]— qj has at most ¡Q + \Q(m—1) — 1 distinct zeros in C, it follows that Q[K] — z1 has at most jq + \Q(m — 1) — 1 distinct zeros in C, and, so, by Corollary 2 and Proposition 2, K reduces to a constant. Using the same arguments as in Case 1, we find that {fj} is normal at z = 0. This completes the induction process and, hence, the proof. □
Acknowledgment. The author is thankful to Prof. N. V. Thin for careful reading of the manuscript and pointing out references [26], [27] and [28]. The author is also thankful to the anonymous referee for his suggestions which improved the presentation of the paper.
References
[1] Aladro G., Krantz S. G. A criterion for normality in Cn. J. Math. Anal. Appl., 1991, vol. 161, pp. 1-8.
DOI: https://doi.org/10.1016/0022-247X(91)90356-5
[2] Beardon A. F., Minda D. Normal families: a geometric perspective. Com-put. Methods Funct. Theory, 2014, vol. 14, pp. 331-355.
DOI: https://doi.org/10.1007/s40315-014-0054-2
[3] Chang J. M. Normality and quasinormality of zero-free meromorphic functions. Acta Math. Sin. Engl. Ser., 2012, vol. 28, no. 4, pp. 707-716. DOI: https://doi.org/10.1007/s10114-011-0297-z
[4] Chang J. M., Xu Y., Yang L. Normal families of meromorphic functions in several variables. Comput. Methods Funct. Theory, 2023.
DOI: https://doi.org/10.1007/s40315-023-00502-7
[5] Charak K. S., Rochon D., Sharma N. Normal families of bicomplex holo-morphic functions. Fractals, 2009, vol. 17, no. 2, pp. 257-268.
DOI: https://doi.org/10.1142/S0218348X09004314
[6] Charak K. S., Rochon D., Sharma N. Normal families of bicomplex mero-morphic functions. Ann. Polon. Math., 2012, vol. 103, pp. 303-317.
[7] Charak K. S., Sharma N. Bicomplex analogue of Zalcman lemma, Complex Anal. Oper. Theory, 2014, vol. 8, no. 2, pp. 449-459.
DOI: https://doi.org/10.1007/s11785-013-0296-4
[8] Chen Q. Y., Yang L., Pang X. C. Normal family and the sequence of omitted functions. Sci. China Math., 2013, vol. 9, pp. 1821-1830.
DOI: https://doi .org/10.1007/s11425-013-4580-6
[9] Chen H. H., Gu Y. X. Improvement of Marty's criterion and its application. Science in China Ser. A, 1993, vol. 36, no. 6, pp. 674-681.
[10] Chen S., Xu Y. Normality relationships between two families concerning shared values. Monatsh. Math., 2021, vol. 196, pp. 1-13.
DOI: https://doi.org/10.1007/s00605-021-01559-z
[11] Deng B. M., Fang M. L., Liu D. Normal families of zero-free meromorphic functions. J. Aust. Math. Soc., 2011, vol. 91, pp. 313--22.
DOI: https://doi.org/10.1017/S1446788711001571
[12] Deng H., Ponnusamy S., Qiao J. Properties of normal harmonic mappings. Monatsh. Math., 2020, vol. 193, no. 3, pp. 605--621.
DOI: https://doi.org/10.1007/s00605-020-01459-8
[13] Deng B. M., Yang D. G., Fang M. L. Normality concerning the sequence of functions. Acta Math. Sin. Eng. Ser., 2017, vol. 33, no. 8, pp. 1154-1162. DOI: https://doi.org/10.1007/s10114-017-6234-z
[14] Dethloff G., Tan T. V., Thin N. V. Normal criteria for families of meromorphic functions. J. Math. Anal. Appl., 2014, vol. 411, no. 2, pp. 675-683. DOI: https://doi.org/10.1016/jomaa.2013.09.059
[15] Fujimoto H. On families of meromorphic maps into the complex projective space. Nagoya Math. J., 1974, vol. 54, pp. 21-51.
DOI: https://doi.org/10.1017/S0027763000024570
[16] Grahl J. Hayman's alternative and normal families of non-vanishing meromorphic functions. Comput. Methods Funct. Theory, 2004, vol. 2, pp. 481-508. DOI: https://doi.org/10.1007/BF03321861
[17] Gu Y. X. A normal criterion of meromorphic families. Sci. Sin., 1979, vol. 1, pp. 267-274.
[18] Hayman W. K. Meromorphic Functions. Clarendon Press, Oxford, 1964.
[19] Hayman W. K. Research Problems in Function Theory. Athlone Press, London, 1967.
[20] Kim K. T., Krantz S. G. Normal families of holomorphic functions and mappings on a Banach space. Expo. Math., 2003, vol. 21, pp. 193-218. DOI: https://doi.org/10.1016/S0723-0869(03)80001-4
[21] Kumar R., Bharti N. Normal families concerning partially shared and proximate values. Sao Paulo J. Math. Sci., 2023, vol. 17, pp. 1076-1085. DOI: https://doi.org/10.1007/s40863-022-00341-9
[22] Montel P. Sur les suites infinies desfonctions. Ann. _École. Norm. Sup., 1907, vol. 24, pp. 233-334.
[23] Montel P. Sur les families de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine. Ann. ¿cole. Norm. Sup., 1912, vol. 29, no. 3, pp. 487-535.
[24] Pang X. C., Zalcman L. Normal families and shared values. Bull. Lond. Math. Soc., 2000, vol. 32, no. 3, pp. 325-331.
DOI: https://doi.org/10.1112/S002460939900644X
[25] Schiff J. L. Normal Families. Springer, Berlin, 1993.
[26] Thin N. V. Normal family of meromorphic functions sharing holomorphic functions and the converse of the Bloch principle. Acta Math. Sci., 2017, vol. 37B, no. 3, pp. 623-656.
DOI: https://doi.org/10.1016/S0252-9602(17)30027-9
[27] Thin N. V. Normal criteria for family meromorphic functions sharing holomorphic function. Bull. Malays. Math. Sci. Soc., 2017, vol. 40, pp. 1413-1442. DOI: https://doi.org/10.1007/s40840-017-0492-x
[28] Thin N. V., Oanh B. T. K. Normality criteria for families of zero-free meromorphic functions. Analysis, 2016, vol. 36, no. 3, pp. 211-222. DOI: https://doi .org/10.1515/anly-2015-0004
[29] Yang L. Normality for families of meromorphic functions. Sci. Sin., 1986, vol. 29, pp. 1263-1274.
[30] Yang L. Value Distribution Theory. Springer-Verlag, Berlin, 1993.
[31] Wu, H. Normal families of holomorphic mappings. Acta Math., 1967, vol. 119, pp. 193-233. DOI: https://doi.org/10.1007/BF02392083
[32] Xie J., Deng B. M. Normality criteria of zero-free meromorphic functions. Chin. Quart. J. Math., 2019, vol. 34, no. 3, pp. 221-231.
[33] Zalcman L. A heuristic principle in complex function theory. Amer. Math. Monthly, 1975, vol. 82, pp. 813-817.
DOI: https://doi.org/10.1080/00029890.1975.11993942
[34] Zalcman L. Normal families: new perspectives. Bull. Amer. Math. Soc., 1998, vol. 35, pp. 215-230.
DOI: https://doi.org/10.1090/S0273-0979-98-00755-1
Received March 06, 2024. In revised form, April 30, 2024. Accepted May 01, 2024. Published online May 26, 2024.
Nikhil Bharti
Department of Mathematics University of Jammu Jammu-180006, India E-mail: nikhilbharti94@gmail.com
