On some classes of harmonic functions with conditions imposed on coefficients and their arguments Текст научной статьи по специальности «Математика»

Trudy Petrozavodskogo gosudarstvennogo universiteta

Seria “Matematika” Vypusk 13, 2006

UDK 517.54

ON SOME CLASSES OF HARMONIC FUNCTIONS WITH CONDITIONS IMPOSED ON COEFFICIENTS AND THEIR ARGUMENTS

Z. J. Jakubowski, A. Lazinska

In this paper we consider a few classes of functions f harmonic in the unit disc A of the form f = h + g, where h, g are suitably normalized functions holomorphic in A. Our special attention is drawn to some classes generated by respective coefficient conditions and to classes of functions with conditions imposed on coefficient arguments. We examine relationships between these conditions and some analytic conditions of stalikeness or convexity of considered functions.

1. Let A = {z € C : |z| < 1}. We consider complex functions harmonic in the disc A of the form

OO OO

f = h + g, h(z) = z + ^ anzn, g(z) = ^ bnzn, z € A, |bi| < 1.

n=2 n=1

(1)

It is known ([1]) that functions f of the form (1) are locally univalent and sense-preserving if and only if

|g'(z)| < |h'(z)|, z € A. (2)

In 1984 J. Clunie and T. Sheil-Small ([2]) published their studies of some geometric properties of univalent functions f of the form (1) that are sense-preserving in A. In paper [2] the authors examined, among others, convexity and close-to-convexity of such functions. They pointed to the

© Z. J. Jakubowski, A. Lazinska, 2006

0Mathematics Subject Classification: 30C45, 30C50, 31A05.

Key Words and Phrases: harmonic functions, starlikness, convexity, coefficient conditions.

fact that if a complex harmonic univalent and sense-preserving function of the form (1) maps the disc A onto a convex domain, then not all images f (Ar), where Ar = {z G C : |z| < r}, r G (0,1), need to be convex. As an example there was given the function fo = ho + g0 of the form

zz

fo(z) = Re----------+ i Im-----,

1 — z (1 — z)2

where

z — 1 z2 1 z2

h0(z) = (1 —2z)2 , g0(z) = — (1 - z)2 , z G A.

We have f0(A) = {w G C :Re w > — 2} and for every r G (%/2 — 1,1) the set f0(Ar) is not convex ([2], Remark 5.6, see also [3], pp. 40-41, 46-48).

It is known that this situation is different from the case of holomorphic univalent functions h, where h(A) is convex if and only if for each r e (0,1) the set h(Ar) is convex.

For holomorphic functions we know (see [4]) coefficient conditions, which imply the univalence of functions and the starlikeness or convexity of the image of A. In consequence, in this case the image of every disc Ar, r G (0, 1), is starlike or convex, respectively.

In many papers we can find studies concerning influence of appropriate coefficient conditions on geometric properties of complex harmonic functions (e.g. [5], [6], [7], [8], [9], [10]). Various authors considered some classes of functions of the form (1) where the signs of coefficients of h and g are fixed (e.g. [11], [12]). We recall some results connected with these problems.

In 1990 Y. Avci and E. Zlotkiewicz proved the following theorems.

Theorem A ([5]). If a function f of the form (1) satisfies the condition

n (|an| + |bn|) < 1 — |b11, (3)

n=2

then f is univalent and sense-preserving in the disc A.

Theorem B ([5]). Let f be a function of the form (1) such that b1 = 0. If

^ n (|a„| + |6„|) < 1,

n=2

then f (A) is a domain starlike with respect to the point 0.

The result contained in Theorem B we can find also in paper [12] published by H. Silverman in 1998. Moreover, H. Silverman considered the case when for a function f of the form (1) we have

an < 0, n = 2, 3,..., bn < 0, n = 1, 2,.... (4)

Among others, he proved

Theorem C ([12]). Let f be a function of the form (1) such that the inequalities (4) hold and bi = 0. The function f is sense-preserving, univalent and maps the disc A onto a domain starlike with respect to the point 0 if and only if it satisfies the condition (3).

In papers [7], [10] and [11] one can find a generalization of Theorems B and C where the restriction bi =0 is omitted.

In the mentioned papers the conclusion that the set f(A) is starlike follows from the starlikeness of f(Ar) for every r e (0,1), i.e. from the fact that if a function f of the form (1) satisfies the condition (3) then

d , i9 , rei0h/(rei0) — reieg/(reie)

— (argf (re ^ =------------ ---- ----- > 0 (5)

h(rei0) + g(rei0)

for any d G (0, 2n), r e (0,1).

Inspirations for these results were the appropriate theorems concerning functions holomorphic in the disc A. Among the oldest papers there should be mentioned e.g. [4], [13], [14], [15], [16], [17], [18], [19].

2. Let VH denote the class of functions f of the form (1) such that b1 G (0,1) and

an = —|an|e-i(n-1)^ , bn = |bn|e-i(n-1)^, n = 2, 3,..., (6)

where if G (0, 2n), if = y>(f).

In 2002 J. M. Jahangiri and H. Silverman ([20]) published some consideration on starlike functions of the class Vh . One of the main theorems of this paper is given below.

Theorem D ([20]). Let f e . The function f is sense-preserving, univalent in A and maps each disc Ar, r e (0,1), onto a starlike domain if and only if the condition (3) holds.

We will prove a slightly more general theorem.

Theorem 1. If a function f of the form (1) belongs to the class and satisfies the condition (2), i.e. is locally univalent and sense-preserving in A, then it satisfies the condition (3).

Proof. Let a function f of the form (1) belong to the class . Then bi e (0,1) and there exists ^ e (0, 2n) such that

h(z) = z - 52 Kl6

-i(n-1)^zn

z = z — e

- eiv^ k|(.

aJ (e ®vz

z A,

n=2

n=2

g(z) = biz + 52 |bn|e-i(n-1)vzn = biz + e^^ |bJ (e—ivz)n , z e A.

n=2 n=2

Hence we obtain

h'(z) = 1 - ^ n|a„|e-i(n-1)vzn-i = 1 - ^ n|aJ (e-i^z)

n—1

z A,

n=2

n=2

/(z) = b1 + ^ n|bn|e—i(n—1)^zn—1 = b1 + ^ n|bn| (e—^z)

n—1

z A.

Assume that the condition (2) holds, i.e. for each z e A we have |h'(z)| > |g'(z)|, so

2

1 - > ' n|an| (e ®vz)

1

>

b1 + 52 n|bn| (e iVz)

1

z A.

In particular, setting z = reiV, where r e (0,1), we obtain

1 - 52 n|anb

n1

>

b1 + 52 n|bn|r

n=2

n1

n

The expressions in the modulus signs are real and the sum on the right-hand side of the above inequality is nonnegative. Therefore

1 — 52 nia«i

n|a„ |rn 1

> bi + 52 n|bn|r” 1 ^ 0, r e (0,1).

n=2

From this fact and according to the continuity of power series we get either

(a) 1 - ^ n|a„|rn-i < —bi - ^ n|6„|rn-i, r e (0, 1),

A>n |/ \ U1 / J n|

n=2 n=2

(b) 1 — ^2 n|an|rn-i > bi + ^2 n|bn|rn-i, r e (0,1).

n=2 n=2

In the case (a) letting r ^ 0+ we obtain bi < —1, which contradicts the assumption that bi e (0,1).

In the case (b) we have

^ n (|an| + |bn|) rn-i < 1 — bi, r e (0, 1).

n=2

Consequently, if r ^ 1-, then we get the condition (3), which completes the proof. □

If f e satisfies the condition (3) then, according to Theorem A and the Lewy’s result ([1]), it satisfies the condition (2). By Theorem 1 we have

Corolary 1. For functions f e the conditions (2) and (3) are equivalent.

Let H* denote the class of functions f of the form (1) satisfying the condition (2), which are univalent in the disc A and map A onto domains starlike with respect to the point 0.

From the presented facts we have the next corollary (see [20]).

or

Corolary 2. Let f e . Then the following conditions are equivalent:

i) f satisfies the condition (2);

ii) f satisfies the condition (3);

iii) f e H*.

The implication iii) ^ i) is a direct consequence of the definition of the class H*. The implication ii) ^ iii) follows from the mentioned results contained in papers [5] and [10].

It is worth mentioning that if a function f of the form (1) belongs to the class Vh and its coefficients are real, then either an < 0 and bn > 0 for n = 2, 3,... (f = 0) or an = ( —1)n|an| and bn = ( — 1)n+i|bn| for n = 2, 3,... (f = n) (see [10]). The class Vh does not contain functions with coefficient satisfying the inequalities (4).

3. We know the following properties of holomorphic functions.

Theorem E ([4]). If a holomorphic function h of the form

h(z) = z + 52 °nzn

n=2

satisfies the condition

then

h(z)

= 0,

'^nWn | < 1,

n=2

zh (z)

h(z)

1

z A,

< 1 for z e A.

(7)

(8) (9)

Corollary A ([4]). If a function h of the form (7) satisfies the condition (8), then h is univalent and starlike in A. Moreover, we have

d

0 < ~dd (arg h(rei0)) < 2 , 0 e (0, 2n), r e (0, 1).

(10)

z

We also have

Theorem F ([19]). If a function h is of the form

+ ^>

h(z) = z — , z € A, a„ > 0, n = 2, 3,.. ., (11)

n=2

and the conditions (9) hold, then we have (8).

Corollary B ([19]). A function h of the form (11) satisfies the condition

(8) if and only if it satisfies (9).

In [12] H. Silverman observed that these properties, especially (9), cannot be directly extended to harmonic functions f = h + g of the form

(1) or respectively such that

+ ^ + ^>

h(z) = z — anzn , g(z) = — bnzn, an, bn > 0, n = 2, 3, .. ..

n=2 n=2

An extention of (9) can be obtained on an additional assumption.

For a function f of the form (1) we put

1 + 52 na„zn-M — z I bi + 52 nb„z”-1 j n=2 / \ n=2 J

(12) for z € A.

It is clear that if Jf denotes the Jacobian of f, i.e. Jf = |h'|2 — |g'|2 for f = h + g, then the condition (2) is equivalent to the condition

Jf (z) > 0 , z € A. (13)

By (12) we have |f*(z)| > |z| (|h'(z)| — |g'(z)|), z € A. We can observe

that if the condition (13) holds, then

|f*(z)| > 0, z € A \{0}.

If f *(zo) =0 for a z0 € A \ {0}, then by (12) we have zoh'(zo) = zog'(zo),

so

f*(zo)=0 =^ Jf (zo) = 0, zo € A \{0}.

Let us note that the converse property does not hold.

f• (z) := zh(z) — zg'(z) = z

Set

z

fi = hi + gl, hi(z) = --------, gi(z) = biz, z € A, |bi| < 1.

1 — z

Then for z € A we have

Jf1 (z)= (iT—^ + |bi|)(— |bi|) , fl‘(z)=(Y^ — biz.

If bi € (4, 1), then there exists exactly one point zbi € A such that

(i_i^)2 = bi, z6i € ( —1,0). Hence

Jfi (zbi) = 0 and f (zbi) = 0.

If I < |bi | < 1, bi = bi, bi € D, where D = w(A), w(z) = (i_z)2, z € A, then there Therefore

then there exists also exactly one point z^ = z^ such that (i_i )2 = bi.

Jfi (zbi) = 0 and f • (zbi) = 0,

which follows from the fact that f • (zbi) = 2i Im(bizbi) = 2i Im ^ (i_Zi )2).

Let f be a function of the form (1). As it has been known (e.g. [20],

[10]), if f (z) =0 for z € A \ {0}, then from (5) and (12) we have

d- (org f (reie)) = Re , (14)

where

f *(reie) 1 — he-™ + E+=°2 (no„ei(n_i)e — n6ne_i(”+i)e) rn-i

f (reie) 1 + 6ie_2ie + ]C+=2 (onei(n_i)e + 6ne_i(”+i)e) rn-i ,

(15)

e € (0, 2n), r € (0, 1).

Theorem 2. If a function f of the form (1) satisfies the condition

+ ^>

E (n|o„| + (n + 2)|b„|) < 2, oi = 1, (16)

then it satisfies the condition (3) and

f *(z)

f(z)

1

< 1 , z € A \{0}. (17)

Proof. Let f be a function of the form (1) satisfying the condition (16). Obviously, then the condition (3) holds and therefore

52 (|on| + |bn|) < 2, oi = 1,

n=i

Consequently, for z € A \ {0} we have

( +TO +TO \

|f-(z)| > |z| 1 — E n|a„||z|”_i — |bi| — E n|b„||z|”_M > 0 ,

|f(z)| > |z| 1 — £ |a„||z|”_i — |bi| — £ |b„||z|n_M > 0 .

\ n=2 n=2 /

Hence

f-(z)f (z) = 0 for z € A \{0}. (18)

According to (16), for z = rei0, r € (0,1), e € (0, 2n), we obtain |f (z)| — |f-(z) — f (z)| =

+

1 + bie_2ie + ^+=°2 (a„ei(n_i)e + b„e_i(n+i)e) r’

—26ie_2ie + E+=°2 ((n — 1)a„ei(n_i)e — (n + 1)6’e_i(n+i)^ rn_i > r (1 — |bi | — E+=°2 (|on| + |bn|) rn_i) +

—r (2|bi| + E+=°2 ((n — 1)|On| + (n + 1)|b„|) rn_i) =

= r (1 — 3|bi| — E+=°2 (n|on| + (n + 2)|bn|) r”_i) > °.

Thus, by (15) and (18), we get (17). □

By (14) and from Theorem 2 we have

>

Corolary 3. If a function f of the form (1) satisfies the condition (16), then

d de

0 <7^ (argf (re")) < 2 , r € (0,1), e € (0, 2n). (19)

In paper [12] H. Silverman gave an example of a function f of the form (1) such that the condition (3) holds for it, but the condition (19) is not satisfied (f2(z) = z — 2z2, z € A).

Let us consider the function f3(z; b) = z + bzn, z € A, b € R, n € {2, 3,...}. If |b| < ’, then the condition (3) holds, so f3 is univalent and sense-preserving and for each r € (0,1) the set f3(Ar; b) is starlike with

f3 (z.b) ____ 1

f3(z;b)

-(n+i)bzn

respect to the origin. From (12) we obtain f (z.b) — 1 = z+bzn z € A \ {0}, and after some computations we conclude that the condition (17) holds only if |b| < n+2. Thus we get

Corolary 4. The function f3(-; b) satisfies the conditions (16) and (17) if and only if |b| < n+2.

It is evident that the mentioned function f2 does not satisfy this assumption.

Next we consider the function f4^; q) of the form

f4(z; q) = z + Vqnzn = z + —±-------- , z € A, |q| < 1.

n^i1 — qz

For |q| < 5_6^ we get

+ TO +TO +TO 3, 1 21 |2

E n|on| + E(n + 2)|bn| = E(n + 2)|q|n = (f_ | |q2 < 1,

n=2 n=i n=i ( |q|)

so then the condition (16) is fulfilled. In consequence, by Theorem 2 and

Corollary 3 the conditions (3), (17) and (19) hold.

Now we turn our attention to the function f5 of the form (1) with

= 0, n = 2, 3,..., bn = — ^„, > 0, n =1, 2,..., ^i < 1, i.e.

+ ^>

f5(z) = z — E ^»z” , z € A, > 0, n =1, 2,..., ^i < 1. (20)

n=i

Assume that the function f5 of the form (20) satisfies the condition

(17). Note that such functions exist (e.g. f5(z) = z — ^z, z € A, 0 <

A < 3). Consequently, the function f5 satisfies the condition (18). Then

3

we have

f5(r)

1 — Ai —52 ^«r”_i , r € (0, 1).

n=2

Observe that limr^0+ f5(r) =1 — A > 0. By the continuity of f5 and from

(18) we conclude that

f5(r) > 0, r € (0, 1). (21)

According to (17) we have

|f5(r)| — |f5(r) — f5(r)| > 0 , r € (0,1).

Hence, by (12), (20) and (21), we obtain

+ ^ +^>

1 — Ai — E Anrn_i > 2Ai + E(n + 1)Anrn_i , r € (0,1),

n=2 n=2

and thus

+ ^>

3Ai + E(n + 2)Anrn_i < 1 , r € (0,1),

n=2

Letting r ^ 1_ we get

+ ^>

^^(n + 2)^n < 1. n=i

Therefore, by Theorem 2, we have

Corolary 5. The function f5 of the form (20) satisfies the condition

(17) if and only if (16) holds for it.

Let o denote the subclass of such that in (6) we have ^ = 0, i.e. the class of functions of the form

+ ^ +^>

f(z) =z — ^2 a«z” + ^2 z € A, bi€ (0, l), (22)

n=2 n=i

r

where an, bn > 0, n = 2, 3,....

Theorem 3. If a function f e 0 satisfies the condition (17), then the condition (3) is fulfilled.

Proof. Let f e VH0 be of the form (22) and satisfy (17). Then we have

(18) for it. Proceeding similar to that for the function f5 gives

1 + b1 + ^^(6n — an)r” 1 >

n=2

v,n—1

— 261 - ((n — 1)an + (n + 1)bn) r"

n=2

hoo +oo

1 + 61 + ^~~l(bn — an)r” 1 > 261 + ((n — 1)an + (n + 1)bn) r” 1

n=2 n=2

for r e (0,1). Therefore

^ n(a„ + bn)rn-1 < 1 - 61 , r e (0, 1).

n=2

From this, letting r ^ 1-, we obtain (3), which completes the proof. □

so

Remark 1. It is easily seen that for holomorphic functions h of the form (7) the conditions (3) and (16) reduce to (8) and the condition

(17) coincides with (9). Of course, the function z ^ ^*£1, z e A, has removeable singularity at the point 0, and so (9) may be considered in the whole disc A.

4. In this section we make some remarks on convex harmonic functions. The following lemma is known.

Lemma A ([3], p. 108). If f = h + g of the form (1) is univalent, sense-preserving starlike in A, and if H and G are holomorphic functions defined by

zH'(z) = h(z), zG'(z) = —g(z), H (0) = G(0) = 0, (23)

then F = H + G is an univalent, sense-preserving function convex in A.

However, the converse to Lemma A is false ([3], p. 110). We know that the convexity of a harmonic function F follows from the condition (see

[5], [10], [12])

where

n2 (|c„| + |dn|) < 1 — |d1^

(24)

n=2

F = H + G, H(z) = z + ^ cnzn, G(z) = ^ dnzn, z e A, |d1| < 1.

n=2 n=1

_ (25)

If f = h + g is of the form (1) and H, G are such that we have (23), then

ncn = a„, ndn = — bn, n =1,2,. .. . (26)

If F of the form (25) satisfies the condition (24), then by (26) we obtain the condition (3) for the function f, and conversely.

From the mentioned facts we obtain

Theorem 4. If a function F of the form (25) satisfies the condition (24), then

(zH'(z))' > (zG'(z))

z e A.

(27)

Proof. Let F = H + G be of the form (25) and let it satisfy (24). Consider the function f = h + g such that the equalities (23) hold. Then by (26) f satisfies (3). From Theorems A, C we conclude that Jf (z) > 0, z e A, which gives

(zH'(z))' > — (zG'(z))' , z e A,

i.e. the condition (27). □

Moreover, we know the following theorem.

Theorem G ([5], [10], [12]). If a function F of the form (25) satisfies the condition (24), then F(Ar) is convex for each r e (0,1).

Consequently, according to Theorems C and G and Lemma A, for harmonic functions satisfying the condition (3) or (24), respectively, we have the "full"Alexander theorem.

Let us return to the class .

Theorem 5. If a function F of the form (25) belongs to the class and satisfies the condition (27), then it satisfies (24).

Proof. From (6) and (25) for a function F G we have

(zH/(z))/ = 1 — n2|cn| (e-iVz)” 1, z G A,

n=2

(zG/(z))/ = |di| + ^n2|d„| (e-iVz)" 1, z G A,

n=2

so by (27) we get

1 — ^ n2|c„|r"-1

n=2

Hence we obtain (24). □

Consequently, by Theorems 4 and 5, we have

Corolary 6. For a function F G the conditions (24) and (27) are equivalent.

Observe that functions f, F of the forms (1), (25), respectively, such that the equalities (26) hold, cannot belong to simultaneously (see

(6)), except for the holomorphic case (i.e. = dn =0, n =1, 2,...).

Let Hc denotes the class of functions F of the form (25) univalent, sense-preserving in A and mapping A onto convex domains.

According to Theorem 5, Theorem G and Corollary 6, we obtain

Corolary 7. If a function F of the form (25) belongs to the class and satisfies the condition (27), then F G Hc.

For a function F of the form (25) we denote

F**(z) := z (zH/(z))/ + z (zG/(z))/ , z G A. (28)

On account of (12) and (28), it is clear that F•• = (F*)*.

From Theorem 2 we obtain

> |di| + ^ n2|d„|r"-1, r G (0,1).

n=2

Theorem 6. If a function F of the form (25) satisfies the condition

^2 n (n|cn| + (n + 2)|d„|) < 2, ci = 1, (29)

n=1

then

F ••(z)

< 1 , z e A \{0}. (30)

F ••(z^1

F •(z)

In consequance, we have

AL-sQ-Lf(reie))) e (0, 2), r e (0, 1), e e(0, 2n). (31)

The last statement follows from the fact (see e.g. [11]) that

A0\

d ( (d ig \\ F"(reie)

ae(“rHdeF(re‘VJ = Re ' re (0'1)1 ee(0'2n).

Obviously, the condition (29) implies (16).

The function /3(z; b) = z + bz”, z e A, n e {2, 3,...}, for |b| < w(w1+2) satisfies the condition (29), and so (30) and (31), as well. If b = |, n = 2, then this function does not satisfy the condition (29).

Some other properties of the operators (12), (28) and their generalizations were considered e.g. in [21], [22].

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Chair of Nonlinear Analysis

Faculty of Mathematics

University of Lodz

ul. S. Banacha 22, 90-238 Lodz, Poland

E-mail: [email protected] lazinska@math. uni.lodz.pl