Classes of Meromorphic Functions Defined by the Hadamard Product

Volume 2010, Article ID 302583,11pages doi:10.1155/2010/302583

Research Article

Classes of Meromorphic Functions Defined by

the Hadamard Product

J. Dziok

Institute of Mathematics, University of Rzesz´ow, ul. Rejtana 16A, 35-310 Rzesz´ow, Poland

Correspondence should be addressed to J. Dziok,[email protected]

Received 2 July 2009; Revised 21 November 2009; Accepted 5 January 2010

Academic Editor: Vladimir Mityushev

Copyrightq2010 J. Dziok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The object of the present paper is to introduce new classes of meromorphic functions with varying argument of coefficients defined by means of the Hadamard productor convolution. Several properties like the coefficients bounds, growth and distortion theorems, radii of starlikeness and convexity, and partial sums are investigated. Some consequences of the main results for well-known classes of meromorphic functions are also pointed out.

1. Introduction

LetMdenote the class of functions which are analytic inDD1, where

Dr {z∈C: 0<|z|< r}, 1.1

with a simple pole in the pointz 0.ByM,we denote the class of functionsf ∈ Mof the form

fz 1 z

∞

n1

anzn z∈ D. 1.2

Also, byTε

η η ∈R, ε∈ {0,1},we denote the class of functionsf ∈ Mof the form1.2for which

Forη0,we obtain the classesT0₀andT1₀of functions with positive coefficients and negative coefficients, respectively.

Motivated by Silverman1, we define the class

Tε_: η∈R

Tε

η. _1.4

It is called the class of functions with varying argument of coefficients.

Letα∈ 0,1, r ∈0,1.A functionf ∈ Mis said to be meromorphically convex of order αinDrif

Re

1zf

_z

fz

<−α z∈ Dr. 1.5

A functionf ∈ Mis said to be meromorphically starlike of orderαinDrif

Re

_zfz

fz

<−α z∈ Dr. 1.6

We denote byMSc_{αthe class of all functionsf ∈ M, which are meromorphically convex} of order α in D and by MS∗α, we denote the class of all functionsf ∈ M,which are meromorphically starlike of orderαinD. We also set

MSc_MSc_{0, MS}∗_MS∗_{0. 1.7}

It is easy to show that for a functionf∈ T0

η,the condition1.6is equivalent to the following:

zf_fzz1<1−α z∈ Dr. 1.8

LetBbe a subclass of the classM. We define the radius of starlikeness of orderαand the radius of convexity of orderαfor the classBby

R∗_αB inf f∈B

sup r∈0,1:f is meromorphically starlike of orderαinDr;

Rc_αB inf f∈B

sup r∈0,1:f is meromorphically convex of orderαinDr, 1.9

respectively.

Let functionsf, gbe analytic inU:D ∪ {0}.We say that the functionfis subordinate to the functiong, and writefz≺gz or simplyf ≺gif there exists a functionωanalytic inU,|ωz| ≤ |z|z∈ U,such that

In particular, ifFis univalent inU, we have the following equivalence:

fz≺Fz⇐⇒f0 F0, fU⊂FU. 1.11

For functionsf, g∈Mof the form

fz ∞

n−1

anzn, gz ∞

n−1

bnzn, 1.12

byf∗gwe denote the Hadamard productor convolutionoffandg, defined by

f∗gz ∞

n−1

anbnzn z∈ D. 1.13

LetA, Bbe real parameters,−1 ≤A < B ≤1,and letϕ, φbe given functions from the classM.

ByWφ, ϕ;A, B,we denote the class of functionsf ∈ Msuch thatϕ∗fz/0 z∈ D and

φ∗fz

ϕ∗fz ≺

1Az

1Bz. 1.14

Moreover, let us define

TWε_{φ, ϕ;A, B:T}ε_{∩ Wφ, ϕ;A, B,}

TWε η

φ, ϕ;A, B:Tε_η∩ Wφ, ϕ;A, B, 1.15

where the functionsϕ, φhave the form

φz 1 z −1

ε1∞

n1

βnzn, ϕz 1 z −1

ε∞

n1

αnzn z∈ D, 1.16

and the sequences{αn},{βn}are nonnegative real, with

αnβn>0 n∈N. 1.17

Moreover, let us put

It is easy to show that

fz∈ TWzφz, zϕz;A, B⇐⇒ −zfz∈ TWφ, ϕ;A, B,

fz∈ TW0_ηzφz, zϕz;A, B⇐⇒ −zfz∈ TW1_ηφ, ϕ;A, B. 1.19

The object of the present paper is to investigate the coefficient estimates, distortion properties and the radii of starlikeness and convexity, and partial sums for the classes of meromorphic functions with varying argument of coefficients. Some remarks depicting consequences of the main results are also mentioned.

2. Coefficients Estimates

First we mention a sufficient condition for functions to belong to the classWφ, ϕ;A, B.

Theorem 2.1. Let{dn}be defined by1.18,−1 ≤ A < B ≤ 1. If a functionf of the form1.2 satisfies the condition

∞

n1

dn|an| ≤B−A, 2.1

thenfbelongs to the classWφ, ϕ;A, B.

Proof. A functionf of the form1.2belongs to the classWε_{φ, ϕ;A, Bif and only if there} exists a functionω,|ωz| ≤ |z|z∈ D,such that

φ∗fz

ϕ∗fz

1Aωz

1Bωz z∈ D, 2.2

or equivalently

zφ∗fz−zϕ∗fz Bzφ∗fz−Azϕ∗fz

<1 z∈ D. 2.3

Thus, it is sufficient to prove that

_z

Indeed, letting|z|r 0< r <1,we have

_z

ϕ∗fz−zφ∗fz−Bzφ∗fz−Azϕ∗fz

∞ n1

βnαnanzn1−B−A− ∞

n1

BβnAαnanzn1

≤∞

n1

βnαn|an|rn1−B−A ∞

n1

BβnAαn|an|rn1

≤∞

n1

dn|an|rn1−B−A<0,

2.5

whencef∈ Wφ, ϕ;A, B.

Theorem 2.2. Let f be a function of the form 1.2, with 1.3. Then f belongs to the class

TWε

ηφ, ϕ;A, Bif and only if the condition2.1holds true.

Proof. In view of Theorem 2.1, we need only to show that each function f from the class TWε

ηφ, ϕ;A, B satisfies the coefficient inequality 2.1. Let f ∈ TWεηφ, ϕ;A, B. Then by 2.3and1.2, we have

_∞

n1

βnαnanzn1

B−A−∞_n1−1εBβnAαnanzn1

<1 z∈ D. 2.6

Therefore, puttingzreiη _{0≤r <1,and applying1.3, we obtain}

_∞

n1

βnαn|an|rn1

B−A−∞_n1BβnAαn|an|rn1 <1. 2.7

It is clear, that the denominator of the left hand said cannot vanish forr ∈ 0,1.Moreover, it is positive forr0,and in consequence forr ∈ 0,1.Thus, by2.7, we have

∞

n1

1Bβn 1Aαn|an|rn1< B−A, 2.8

which, upon lettingr → 1−, readily yields the assertion2.1.

FromTheorem 2.2, we obtain coefficients estimates for the classTWε

ηφ, ϕ;A, B.

Corollary 2.3. If a functionfof the form1.2belongs to the classTWε

ηφ, ϕ;A, B,then

|an| ≤B−A

dn

where{dn}is defined by1.18. The result is sharp. The functionsfn,ηof the form

fn,ηz 1 z

B−A

ei{n1η−επ}_dnzn z∈ D; n∈N 2.10

are the extremal functions.

3. Distortion Theorems

FromTheorem 2.2, we have the following lemma.

Lemma 3.1. Let a functionf of the form1.2belong to the classTWε

ηφ, ϕ;A, B.If the sequence

{dn}defined by1.18satisfies the inequality

d1≤dn n∈N, 3.1

then

∞

n1

an≤ B−A

d1

. 3.2

Moreover, if

nd1 ≤dn n∈N, 3.3

then

∞

n1

nan≤ B−A

d1

. 3.4

Theorem 3.2. Let a functionf belong to the classTWε

ηφ, ϕ;A, B.If the sequence{dn}defined by 1.18satisfies3.1, then

1 r −

B−A d1 r≤

_fz≤ 1 r

B−A

d1 r |z|r <1. 3.5

Moreover, if3.3holds, then

1 r2 −

B−A d1 ≤

fz≤ 1 r2

B−A d1 |

z|r <1. 3.6

Proof. Let a functionfof the form1.2belong to the classTWε

ηφ, ϕ;A, B, |z|r <1.Since

fz1 z

∞

n1

anzn

≤ 1 r ∞ n1 |an|rn

1 r r

∞

n1

|an|rn−1≤ 1 r r

∞

n1 |an|,

fzz ∞

n1

anzn

≥ 1 r − ∞ n1 |an|rn

1 r −r

∞

n1

|an|rn−1≥ 1 r −r

∞

n1 |an|,

3.7

then byLemma 3.1we have3.5. Analogously we prove3.6.

4. The Radii of Convexity and Starlikeness

Theorem 4.1. The radius of starlikeness of orderαfor the classTW0

ηφ, ϕ;A, Bis given by

R∗_αTW0_ηφ, ϕ;A, B inf n∈N

1−αdn n−αB−A

1/n1

, 4.1

wherednis defined by1.18.

Proof. The functionf ∈ T0

ηof the form1.2is meromorphically starlike of orderαin the disk

Dr,0< r≤1,if and only if it satisfies the condition1.8. Since

zf_fzz1

_∞

n1n1 anzn1 1∞_n1anzn1

≤

_∞

n1n1|an||z|n1

1−∞_n1|an||z|n1 , 4.2

putting|z|r,the condition1.8is true if

∞

n1

n−α 1−α|an|r

n1 _{≤1. 4.3}

ByTheorem 2.2, we have

∞

n1

dn

B−A|an| ≤1. 4.4

Thus, the condition4.3is true if

n−α 1−αr

n1 _≤ dn

that is, if

r ≤

1−αdn n−αB−A

1/n1

n∈N. 4.6

It follows that each functionf ∈ TW0

ηφ, ϕ;A, Bis meromorphically starlike of orderαin the disk Dr, where r R∗TW0

ηφ, ϕ;A, B is defined by4.1. Moreover, the radius of starlikeness of the functionsfn,ηdefined by2.10is given by

r_n∗min

1,

1−αdn n−αB−A

1/n1

n∈N. 4.7

Thus we have4.1.

Theorem 4.2. The radius of convexity of orderαfor the classTW1

ηφ, ϕ;A, Bis given by

Rc_αTW1_ηφ, ϕ;A, Binf n∈N

1−αdn nn−αB−A

1/n1

, 4.8

wherednis defined by1.18.

Proof. The proof is analogous to that ofTheorem 4.1, and we omit the details.

5. Partial Sums

Letf∈ Mbe a function of the form1.2. Motivated by Silverman2and Silvia3 see also

4, we define the partial sumsfmdefined by

fmz 1 z

m

n1

anzn m∈N. 5.1

In this section, we consider partial sums of functions from the class TWε

ηφ, ϕ;A, B and obtain sharp lower bounds for the real part of ratios offtofmandftofm.

Theorem 5.1. Letm∈Nand let the sequence{dn},defined by1.18, satisfy the inequalities

B−A≤dn≤dn1 m∈N. 5.2

If a functionfbelongs to the classTWε

ηφ, ϕ;A, B,then

Re _fz

fmz

≥1−B−A d_m1

z∈ D, 5.3

Re

_f

mz fz

≥ dm1

B−Adm1

z∈ D. 5.4

Proof. Let a functionfof the form1.2belong to the classTWε

ηφ, ϕ;A, B.Then by5.2and Theorem 2.2, we have

m

n1 |an|

dm1

B−A ∞

nm1 |an| ≤

∞

n1

dn

B−A|an| ≤1. 5.5

If we put

gz dm1 B−A

_fz

fmz−

1−B−A d_m1

1dm1/B−A

_∞

nm1anzn1 1m_n1anzn1

z∈ D,

5.6

then it suffices to show that

Regz≥0 z∈ D, 5.7

or

g_gz_z−1₁≤1 z∈ D. 5.8

Applying5.5, we find that

g_gz_z−1₁≤ dm1/B−A∞nm1|an| 2−2n_n1|an| −dm1/B−A

_∞

nm1|an|

≤1 z∈ D, 5.9

which readily yields the assertion5.3. In order to see thatffm1,η gives the result sharp, we observe that forzreiη_{we have}

fz fmz

1−B−Arm 2

d_m1

r_−→→1−

1−B−A d_m1

. 5.10

Similarly, if we take

hz B−Adm1 _fmz

fz −

dm1

B−Adm1

z∈ D, 5.11

and make use of5.5, we can deduce that

h_hz_z−1₁≤ 1 dm1/B−A

_∞

nm1|an|

2−2m_n1|an| −1−dm1/B−A∞_nm1|an|

≤1 z∈ D, 5.12

Theorem 5.2. Letm∈Nand let the sequence{dn},defined by1.18, satisfy the inequalities5.2. If a functionfbelongs to the classTWε

ηφ, ϕ;A, B,then

Re

fz fmz

≥1−B−Am1

dm1 z∈ D,

Re

_f

mz fz

≥ dm1

B−Am1 dm1 z∈ D.

5.13

The bounds are sharp, with the extremal functionfm1,η of the form2.10.

Proof. The proof is analogous to that ofTheorem 5.1, and we omit the details.

Remark 5.3. We observe that the obtained results are true if we replace the class

TWε

ηφ, ϕ;A, BbyTWεφ, ϕ;A, B.

6. Concluding Remarks

We conclude this paper by observing that, in view of the subordination relation1.18, by choosing the functionsφ and ϕ, we can define new classes of functions. In particular, the class

Wϕ;A, B:W−zϕz, ϕz;A, B 6.1

contains functionsf ∈ M,such that

−z

ϕ∗fz

ϕ∗fz ≺

1Az

1Bz. 6.2

A functionf ∈ Mbelongs to the class

Wϕ;α:Wϕ; 2α−1, α 0≤α <1 6.3

if it satisfies the condition

Re

−z

ϕ∗fz

ϕ∗fz

> α z∈ D. 6.4

The classWϕ;αis related to the class of starlike function of orderα. In particular, we have the following relationships:

S∗_{α W} 1 z1−z;α

,

Sc_{α W}

1−2z z1−z2;α

.

Letλbe a convex parameter. A functionf∈ Mbelongs to the class

Wλ

ϕ;A, B:W

λzϕz λ−1z2ϕz,1 z;A, B

6.6

if it satisfies the condition

λzϕ∗fz λ−1z2ϕ∗fz≺ 1Az

1Bz. 6.7

The classes Wϕ;A, B,Wϕ;α,and Wλϕ;A, Bgeneralize well-known important classes, which were investigated in earlier works; see for example5–10.

If we apply the results presented in this paper to the classes discussed above, we can obtain several additional results. Some of these results were obtained in earlier works; see for example5–10.

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