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36Труды Петрозаводского государственного университета
Серия “Математика” Выпуск 11, 2004
YAK 517.54
ON SOME HARMONIC FUNCTIONS RELATED TO HOLOMORPHIC FUNCTIONS WITH A POSITIVE REAL PART
In the paper we examine some holomorphic functions and complex harmonic functions, which satisfy certain conditions of a Mo-canu kind. We also consider their relations with appropriate coefficient conditions. The paper is a natural supplement to the results published in [1] and [2].
A. Let us first consider functions / holomorphic in the unit disc A = = {zEC:|z|<l} and such that /(0) = /'(0) — 1 = 0, i. e. functions of
For a fixed number a E (0,1) by J(a) we denote the class of all functions / of the form (1) satisfying the condition
Remark 1. Some properties of the class J(a), a E (0,1), were examined in 1977 by P. N. Chichra [2]. In 1915 J. W. Alexander [3] proved that J{0) is a class of univalent functions (see also the Noshiro — Warszawski lemma, 1935). The class J{0) was examined by T. H. MacGregor [4] and others as well.
Z. J. Jakubowski, A. Lazinska
the form
(i)
71 = 2
f (z)
Re а----------1-(1 — a)f'(z) >0, z E A.
z
(2)
© Z. J. Jakubowski, A. Lazinska, 2004
Observe that the identity function belongs to every class J(a), ol E (0,1).
Directly from the definition of the class J(a), a E (0,1), we get Proposition 1. Let a E (0,1). If f E J(a) then functions
z r^firz), z e-^fi^z), z E A, r E (0,1), t E M,
also belong to J (a).
Let p denote the known class of Caratheodory functions with a positive real part, i. e. the functions p holomorphic in A such that p(0) = 1 and Re p(z) > 0, z E A.
From the definitions of the classes p, J(a), a E (0,1), we immediately obtain
Proposition 2. Let a G (0,1). If f E J(a), then the function p of the form
p{z)=a^- + (l-a)f(z), z £ A, (3)
z
belongs to the class p. Conversely, if p E p, then the function f of the form (1), which is a solution of the equation (3), belongs to J(a).
Example 1. Let k E N, k > 2, a E (0,1). Consider the functions f(z,k) of the form
f(z,k) = z + akzk, 0 <ak< —— ------------------------—, z £ A. (4)
a + (1 — a)k
It is easy to check that every function f(z,k) of the form (4) belongs to J(a).
We shall prove the next theorem.
Theorem 1. Let a E (0,1). If a holomorphic function f of the form (1) satisfies the condition
+ oo
^ (a+(1 - a)n) |o„| < 1, (5)
71 = 2
then f E J(a).
Proof. Assume that for a E (0,1) a function / of the form (1) satisfies (5). It suffices to show that oi^p- + (1 — a)f'(z) — 1 < 1, z E A. By (1) and (5) we obtain
+ oo
(a + (1 — a)n)
a^—^~ + (1 — a)f'(z) — 1
,71—1
<
+ 00
< ^2 (a + (1 - a)n) |a„||z" 11 < 1, z£ A,
71 = 2
hence / E J(a). □
Remark 2. Every function of the form (4) satisfies the condition (5), of course.
Let T denote the class of functions of the form (1) such that
\an\ <1, n = 2,3,....
It is clear that the known class Sc of convex functions is a subclass of T.
Theorem 1 and the definition of T imply
Corolary 1. If ip(z) = z -h cnzU> z ^ ^ is a function of the
class T and f of the form (1) satisfies the condition (5) for a E (0,1), then the Hadamard product
+ oo
(f * ip)(z) = z+ ^2ancnzn, z e A,
71 = 2
belongs to J(a).
Example 2. Fix a E (0,1) and denote (see [2])
+ oo
fo{z) — Z +
2zr
a + (1 — a)n ’
A.
(6)
71 = 2
The function /o of the form (6) is holomorphic in A and it is suitably normalized. Moreover, for any z E A we have
which gives /0 E J(a). However, we observe that /0 of the form (6) does not satisfy (5).
It appears that with some additional assumptions the condition (5) is not only sufficient but also necessary for a function to belong to J(a).
Let J~(a), a E (0,1), denote the class of functions / E J(a) which are of the form
+00
f(z) = Z-^2\an\zn, z e A, (7)
71 = 2
(see [5], [6]). We have
Theorem 2. Let a E (0,1). A holomorphic function f of the form (7) belongs to the class J~(a) if and only if it satisfies (5).
Proof. If / of the form (7) belongs to J~(a), a E (0,1), then for any x E (0,1) we have
0 < Re
= 1 — (ck + (1 — a)n)
an\xn
71 = 2
Therefore we obtain (5).
The converse statement follows from Theorem 1. □
Corolary 2. Let a e (0,1). If f E J~(a) and
+00
<p(z) = z + ^2 \cn\zn, z E A,
71 = 2
is a function of T, then f * ip E J~ (a).
It is known [7] that if a holomorphic function / of the form (1) satisfies the condition
+00
^n|a„|<l, (8)
71 = 2
then / is univalent and starlike in A.
Denote Ar = {z E C: \z\ < r} for r > 0, with Ai = A. In the paper [2] we can find the theorem on the disc where all functions of the class J(a), a E (0,1), are univalent. The radius of this disc is a solution of an equation. It appears that in the class of functions satisfying (5) we have the next theorem.
Theorem 3. Let a E (0,1) and r* = 1 — f. If a function f of the form (1) satisfies the condition (5), then it is univalent and starlike in Ar#. The constant 1 — f is the best possible.
Proof. The proof is based on Proposition 1, Theorem 1, the known inequality [1]
satisfies the condition (5) and /'(1 — f) = 0. □
B. Let us next consider complex functions harmonic in the disc A of the form
f = h + g, h(z) = z + ^2 anzn, g(z) = ^ 6„z", z € A. (9)
If / is a function of the form (9), then F = h + g is a function holomorphic in A and F(0) = F'(0) - 1 = 0.
For an arbitrarily fixed a E (0,1) by Jh(oi) we denote the class of functions / of the form (9) and such that the function F = h + g belongs to J(a). It means that the function / of the form (9) belongs to the class Jh{ol), ol E (0,1), if and only if
Obviously, J(a) C Jh{ol), a e (0,1).
According to the definition of the class a E (0,1), and Theo-
rem 1 we obtain the theorem.
Theorem 4. Let a E (0,1). If a harmonic function f of the form (9) satisfies the condition
nrn 1 < a + (1 - a)n, a E (0,1), rE(0,r*), n — 2,3,...,
and the condition (8).
The result is sharp because the function /* of the form
z
f*(z) = z-~---------------, zei,
I — OL
n—2
n—2
+ (1 - a) (h'(z) + g'{z)) >0, z G A. (10)
71 = 2
then / E Jh{ol).
Let Tu stand for the class of functions x = (/? + -0, where
+00 +00
ip(z) = z + 'Y^cnzn, i;(z) = '£dnzn, z e A,
n=2 n=2
and \cn| < 1, |dn| < 1, n = 2,3,...
From the definition of the class Tu and by Theorem 4 we obtain:
Corolary 3. Jfx = (/? + '0£ ^ and f of the form (9) satisfies the condition (11), then the Hadamard product of harmonic functions
(/*X)0) = h*<p + g*ip
satisfies (10) and consequently belongs to the class Jh(oi).
Denote by ol E (0,1), the class of functions / E Jh{ol) which
are of the form
+00 +00
f = h + g, h(z) = z-^2\an\zn, g(z) = ~^\bn\zn, z e A. (12)
71 = 2 71 = 2
By the definition of J^(a), a E (0,1), and in view of Theorems 1, 2, 4 and the condition (11) we get the theorem.
Theorem 5. Let a E (0,1). A harmonic function f of the form (12) belongs to the class J^(a) if and only if it satisfies (11).
Remark 3. We can consider harmonic functions x — ^ + V’? where
+ oo +oo
tp(z) = z+ ^2\cn\zn, tp(z) = '^2\dn\zn, z £ A,
71 = 2 71 = 2
|C„|<1, KI < 1, n — 2, 3,... and their Hadamard products with functions f of the form (12). Then we obtain a corollary analogous to Corollary 2.
In the paper [8] we can find the sufficient condition for harmonic functions to be starlike. Applying this theorem, which generalizes the man-tioned result of A. W. Goodman ([7]), we can prove the theorem.
Theorem 6. Let a E (0,1). If a harmonic function f of the form (9) satisfies the condition (11), then f is univalent, sens-preserving and star-like in every disc Ar, where r E (0, r*), r* = 1 — f. The constant 1 — f is the best possible.
Theorems 2, 3, 5, 6 imply
Corolary 4. The univalence and starlikeness radius for the classes J~(a), J]j(a), a E (0,1), is equal to r* = 1 — f.
We can show
Lemma 1. Let a E (0,1), I > 1. Then for any r E (0, ^=7-) and n = 2,3,... we have
nzrn_1 < a + (1 — a)n.
The above inequality is a kind of generalization of the inequality (see [1]) applied in the proof of Theorem 3 and follows from it.
Applying the known from [8] condition concerning convexity of harmonic functions we get the theorem.
Theorem 7. Let a E (0,1). If a harmonic function f of the form (9) satisfies the condition (11), then for any r E (0,rc), rc = ‘^L, the set /(Ar) is a convex domain. The constant is the best possible.
Proof. Fix a E (0,1), r E (0,rc) and assume that / of the form (9) satisfies the condition (11). By Theorem 4 the function / belongs to Jh(ol). Consider the function fr(z) = r~1f(rz), z E A. We have
+00 +00
fr(z) = z + Y, anrn~1zn + J2 Krn~1zn, z e A.
71 = 2 71 = 2
Using Lemma 1 (for I = 2) and (11) we obtain
+00 +00
1>2 {\anrn~1\ + |6„rn_1|) < ^2 (a + (1 “ a)n) (la«l + IM) - < !•
71 = 2 71 = 2
Hence /(Ar), r E (0,rc), is a convex domain (see [8]).
The constant cannot be improved because of e. g. the mentioned function /* E J~(a) C Jh{ol) C Jh(ol). It follows from the known fact that a holomorphic function of tht form z i-» z + azn is convex in A if and only if \a\ < ^2, n E N, n > 2. □
Theorems 5 and 7 give:
Corolary 5. The convexity radius for the classes J~(a), J^(a), a E E (0,1), is equal to rc =
From Proposition 2, the inequality (10) and the known coefficient estimates for functions of the class p we obtain
Proposition 3. Let a E (0,1). If a function f of the form (9) belongs to the class Jh(oi), then
IK| - IMI < —TiT-----------n = 2,3,... (13)
a + (1 — a)n
The estimates (13) are sharp.
The sharpness of the estimates (13) we can observe, among others, for the function /o of the form (6), where an = a+^_a^n, bn = 0, n = 2, 3,...
C. For a fixed a E (0,1) denote by K(a) the class of holomorphic functions / of the form (1) such that
Ite [/'(*)+ (1-«)/"(*)] >0, z e A. (14)
The class K(a), a E (0,1), was partially examined in the paper [2]. The classes J(a), K(a), a E (0,1), are closely related by the following theorem.
Theorem A [2]. Let a E (0,1). If / E K(a), then the function (p of the form <p(z) = zf'(z), z e A, belongs to the class J(a). Conversely if (f E J(a), then the function f of the form f(z) = fj z E A,
belongs to K (a).
By Theorems 1 and A we get the theorem.
Theorem 8. If a holomorphic function / of the form (1) satisfies the condition
+ oo
y (an + (1 - a)n2) \an\ < 1 (15)
71 = 2
for a fixed a E (0,1), then f E K(a).
Remark 4. (i) We can also consider the Hadamard products of functions f of the form (1) satisfying the condition (15) with functions of the class T.
(ii) The condition (15) is not necessary for a function f to belong to the class K(a), a E (0,1). It follows from Example 2 and Theorem A.
Next denote by K~(a), a E (0,1), the class of functions / of the form (7) which belong to K(a). As in the class J~(a), a E E (0,1), we obtain the theorem.
Theorem 9. A holomorphic function f of the form (7) belongs to the class K~(a), a E (0,1), if and only if f satisfies the condition (15).
Let now Kff(a), a E (0,1), stand for the class of harmonic functions of the form (9) (/ = h + g) such that the holomorphic functions F = h + g belong to K(a). Moreover, K^(a) denotes the subclass of Kh{ol), a E (0,1), of functions / of the form (12).
According to the above definition a function / of the form (9) belongs to the class Kh((x), ol E (0,1), if and only if
Re [h'(z) + g'(z) + (1 - a) (h"(z) + g"{z))\ >0, z E A. (16)
We see at ones that K(a) C Kh{ol), a E (0,1).
From the definitions of the considered classes and by Theorem A we obtain the theorem.
Theorem 10. Let a E (0,1). If f E Kh{ol), then the function ip of the form (p(z) = zh'(z) + zg'(z), z E A, belongs to the class Jh(oi). Conversely if a function cp = H + G is in the class Jh{ol), then the
function / of the form f(z) = ^p~d( + z E A, belongs to
the class Kh{ol).
Theorems 8, 9, 10 and the condition (16) give the following two theorems.
Theorem 11. If a harmonic function f of the form (9) satisfies the condition
+ oo
^ (an + (1 - a)n2) (|a„| + |6n|) < 1 (17)
71 = 2
for a fixed a E (0,1), then f E Kh(oi).
Theorem 12. A harmonic function / of the form (12) belongs to the class K]j(a), a E (0,1), if and only if f satisfies the condition (17).
Remark 5. The class of harmonic functions of the form (9) and satisfying the condition (17) for a fixed a E (0,1), was considered in the paper
[1]. The class Kh(oi), a E (0,1), is a certain generalization of this class. By Remark 4(H) we observe that these classes are not equal. It seems that the above presented theorems form a natural supplement to the results contained in the article [1].
References
[1] Lazinska A. On complex mappings in the unit disc with some coefficient conditions / A. Lazinska // Folia Sci. Univ. Techn. Resoviensis. 2002. V. 199(26). R 107-116.
[2] Chichra P. N. New subclasses of the class of dose-to-convex functions / P. N. Chichra // Proc. Amer. Math. Soc. 1977. V. 62(1). P. 37-43.
[3] Alexander J. W. Functions which map the interior of the unit circle upon simple regions / J. V. Alexander // Annals of Math. 1915. V. 17. P. 12-22.
[4] MacGregor T. H. Functions whose derivative has a positive real part / T. H. MacGregor // Trans. Amer. Math. Soc. 1962. V. 104. P. 532-537.
[5] Silverman H. Harmonic univalent functions with negative coefficients / H. Silverman // J. Math. Anal, and Appl. 1998. V. 220. P. 283-289.
[6] Silverman H. Subclasses of harmonic univalent functions / H. Silverman, E. M. Silvia // New Zealand J. Math. 1999. V. 28. P. 275-284.
[7] Goodman A. W. Univalent functions and nonanalytic curves / A. W. Goodman // Proc. Amer. Math. Soc. 1957. V. 8(3). P. 598-601.
[8] Avci Y. On harmonic univalent mappings / Y. Avci, E. Zlotkiewicz // Ann. Univ. Mariae Curie-Sklodowska. Sec. A. 1990. V. XLIV(l). P. 1-7.
Chair of Special Fuctions Faculty of Mathematics University of Lodz,
ul. S. Banacha 22, 90-238 Lodz, Poland E-mail: [email protected] [email protected]
