International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 915625 | 10 pages | https://doi.org/10.1155/2012/915625

Subclass of Multivalent Harmonic Functions with Missing Coefficients

Academic Editor: Attila Gilányi
Received20 Mar 2012
Accepted08 Jul 2012
Published12 Aug 2012

Abstract

We have studied subclass of multivalent harmonic functions with missing coefficients in the open unit disc and obtained the basic properties such as coefficient characterization and distortion theorem, extreme points, and convolution.

1. Introduction

A continuous function 𝑓=𝑢+𝑖𝑣 is a complex-valued harmonic function in a simply connected complex domain 𝐷⊂ℂ if both 𝑢 and 𝑣 are real harmonic in 𝐷. It was shown by Clunie and Sheil-Small [1] that such harmonic function can be represented by 𝑓=ℎ+𝑔, where ℎ and 𝑔 are analytic in 𝐷. Also, a necessary and sufficient condition for 𝑓 to be locally univalent and sense preserving in 𝐷 is that |ℎ(𝑧)|>|ğ‘”î…ž(𝑧)| (see also, [2–4]).

Denote by 𝐻 the family of functions 𝑓=ℎ+𝑔, which are harmonic univalent and sense-preserving in the open-unit disc 𝑈={𝑧∈ℂ∶|𝑧|<1} with normalization 𝑓(0)=ℎ(0)=ğ‘“î…žğ‘§(0)−1=0.

For 𝑚≥1,0≤𝛽<1,and𝛾≥0,let 𝑅(𝑚,𝛽,𝛾)denote the class of all multivalent harmonic functions 𝑓=ℎ+𝑔 with missing coefficients that are sense-preserving in 𝑈, and ℎ,𝑔 are of the form ℎ(𝑧)=𝑧𝑚+âˆžî“ğ‘›=𝑚+1ğ‘Žğ‘›+1𝑧𝑛+1,𝑔(𝑧)=âˆžî“ğ‘›=𝑚𝑏𝑛+1𝑧𝑛+1(𝑚≥1;𝑧∈𝑈)(1.1) and satisfying the following condition: Re1+ğ›¾ğ‘’ğ‘–ğœ™î€¸ğ‘§ğ‘“î…ž(𝑧)ğ‘§î…žğ‘“(𝑧)−𝛾𝑚𝑒𝑖𝜙≥𝑚𝛽(𝑚≥1;0≤𝛽<1;𝛾≥0;𝜙real),(1.2) where ğ‘§î…ž=𝜕𝜕𝜃𝑧=𝑟𝑒𝑖𝜃,ğ‘“î…žğœ•(𝑧)=𝑓𝜕𝜃𝑟𝑒𝑖𝜃.(1.3)

We note that:(i)𝑅(𝑚,𝛽,1)=𝑅(𝑚,𝛽) with ğ‘Žğ‘š+1;𝑏𝑚≠0 (see Jahangiri et al. [5]);(ii)𝑅(1,𝛽,𝛾)=𝐽𝐻(𝛼,𝛽,𝛾) (see Kharinar and More [6]);(iii)𝑅(1,𝛽,1)=𝐺𝐻(𝛽)with ğ‘Ž2;𝑏1≠0 (see Rosy et al. [7] and Ahuja and Jahangiri [2]).

Also, the subclass denoted by 𝑇(𝑚,𝛽,𝛾) consists of harmonic functions 𝑓=ℎ+𝑔, so that ℎ and 𝑔 are of the form ℎ(𝑧)=ğ‘§ğ‘šâˆ’âˆžî“ğ‘›=𝑚+1ğ‘Žğ‘›+1𝑧𝑛+1,𝑔(𝑧)=âˆžî“ğ‘›=𝑚𝑏𝑛+1𝑧𝑛+1î€·ğ‘Žğ‘›+1;𝑏𝑛+1.≥0;𝑚≥1;𝑧∈𝑈(1.4)

We note that:(i)𝑇(𝑚,𝛽,1)=𝑇(𝑚,𝛽) with ğ‘Žğ‘š+1;𝑏𝑚≠0(see Jahangiri et al. [5]);(ii)𝑇(1,𝛽,𝛾)=𝐽𝐻(𝛼,𝛽,𝛾)(see Kharinar and More [6]);(iii)𝑇(1,𝛽,1)=𝐺𝐻(𝛽) with ğ‘Ž2;𝑏1≠0(see Rosy et al. [7] and Ahuja and Jahangiri [2]).

From Ahuja and Jahangiri [2] with slight modification and among other things proved, if 𝑓=ℎ+𝑔is of the form (1.1) and satisfies the coefficient condition âˆžî“ğ‘›=𝑚−1(𝑛+1)−𝑚𝛽||ğ‘Žğ‘š(1−𝛽)𝑛+1||+(𝑛+1)+𝑚𝛽||𝑏𝑚(1−𝛽)𝑛+1||î‚¹î€·ğ‘Žâ‰¤2𝑚=1;ğ‘Žğ‘š+1=𝑏𝑚,=0(1.5) then the harmonic function 𝑓is sense-preserving, harmonic multivalent with missing coefficients and starlike of order 𝛽(0≤𝛽<1) in 𝑈.They also proved that the condition (1.5) is also necessary for the starlikeness of function 𝑓=ℎ+𝑔of the form (1.4).

In this paper, we obtain sufficient coefficient bounds for functions in the class 𝑅(𝑚,𝛽,𝛾). These sufficient coefficient conditions are shown to be also necessary for functions in the class 𝑇(𝑚,𝛽,𝛾). Basic properties such as distortion theorem, extreme points, and convolution for the class 𝑇(𝑚,𝛽,𝛾) are also obtained.

2. Coefficient Characterization and Distortion Theorem

Unless otherwise mentioned, we assume throughout this paper that 𝑚≥1,0≤𝛽<1,𝛾≥0,and𝜙is real. We begin with a sufficient condition for functions in the class 𝑅(𝑚,𝛽,𝛾).

Theorem 2.1. Let 𝑓=ℎ+𝑔 be such that ℎ and 𝑔 are given by (1.1). Furthermore, let âˆžî“ğ‘›=𝑚−1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)||ğ‘Žğ‘š(1−𝛽)𝑛+1||+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)||𝑏𝑚(1−𝛽)𝑛+1||≤2,(2.1) where ğ‘Žğ‘š=1andğ‘Žğ‘š+1=𝑏𝑚=0.Then 𝑓 is sense-preserving, harmonic multivalent in 𝑈 and 𝑓∈𝑅(𝑚,𝛽,𝛾).

Proof. To prove 𝑓∈𝑅(𝑚,𝛽,𝛾), by definition, we only need to show that the condition (2.1) holds for 𝑓. Substituting ℎ+𝑔for 𝑓 in (1.2), it suffices to show that ⎧⎪⎨⎪⎩Re1+ğ›¾ğ‘’ğ‘–ğœƒî€¸î‚€ğ‘§â„Žî…ž(𝑧)âˆ’ğ‘§ğ‘”î…žî‚î€·(𝑧)−𝑚𝛽+ğ›¾ğ‘’ğ‘–ğœƒî€¸î‚€â„Ž(𝑧)+𝑔(𝑧)ℎ(𝑧)+âŽ«âŽªâŽ¬âŽªâŽ­ğ‘”(𝑧)≥0,(2.2) where ℎ(𝑧)=(𝜕/𝜕𝑧)ℎ(𝑧) and ğ‘”î…ž(𝑧)=(𝜕/𝜕𝑧)𝑔(𝑧). Substituting for ℎ,𝑔,ℎ, and ğ‘”î…ž in (2.2), and dividing by 𝑚(1−𝛽)𝑧𝑚, we obtain Re(𝐴(𝑧)/𝐵(𝑧))≥0, where 𝐴(𝑧)=1+âˆžî“ğ‘›=𝑚+1(𝑛+1)1+𝛾𝑒𝑖𝜃−𝑚𝛽+ğ›¾ğ‘’ğ‘–ğœƒî€¸ğ‘Žğ‘š(1−𝛽)𝑛+1𝑧𝑛−𝑚+1âˆ’î‚µğ‘§ğ‘§î‚¶ğ‘šâˆžî“ğ‘›=𝑚+1(𝑛+1)1+𝛾𝑒−𝑖𝜃+𝑚𝛽+𝛾𝑒−𝑖𝜃𝑏𝑚(1−𝛽)𝑛+1𝑧𝑛−𝑚+1,𝐵(𝑧)=1+âˆžî“ğ‘›=𝑚+1ğ‘Žğ‘›+1𝑧𝑛−𝑚+1+î‚µğ‘§ğ‘§î‚¶ğ‘šâˆžî“ğ‘›=𝑚+1𝑏𝑛+1𝑧𝑛−𝑚+1.(2.3) Using the fact that Re(𝑤)≥0 if and only if |1+𝑤|>|1−𝑤| in 𝑈, it suffices to show that|𝐴(𝑧)+𝐵(𝑧)|−|𝐴(𝑧)−𝐵(𝑧)|≥0.Substituting for 𝐴(𝑧) and 𝐵(𝑧) gives ||||−||||=|||||𝐴(𝑧)+𝐵(𝑧)𝐴(𝑧)−𝐵(𝑧)2+âˆžî“ğ‘›=𝑚+1(𝑛+1)1+𝛾𝑒𝑖𝜃−𝑚−1+2𝛽+ğ›¾ğ‘’ğ‘–ğœƒî€¸ğ‘Žğ‘š(1−𝛽)𝑛+1𝑧𝑛−𝑚+1âˆ’î‚µğ‘§ğ‘§î‚¶ğ‘šâˆžî“ğ‘›=𝑚(𝑛+1)1+𝛾𝑒−𝑖𝜃+𝑚−1+2𝛽+𝛾𝑒−𝑖𝜃𝑏𝑚(1−𝛽)𝑛+1𝑧𝑛−𝑚+1|||||−|||||âˆžî“ğ‘›=𝑚+1(𝑛+1)1+𝛾𝑒𝑖𝜃−𝑚1+ğ›¾ğ‘’ğ‘–ğœƒî€¸ğ‘šğ‘Ž(1−𝛽)𝑛+1𝑧𝑛−𝑚+1âˆ’î‚µğ‘§ğ‘§î‚¶âˆžî“ğ‘›=𝑚(𝑛+1)1+𝛾𝑒−𝑖𝜃+𝑚1+𝛾𝑒−𝑖𝜃𝑚𝑏(1−𝛽)𝑛+1𝑧𝑛−𝑚+1|||||≥2âˆ’âˆžî“ğ‘›=𝑚+1(𝑛+1)(1+𝛾)−𝑚(2𝛽+𝛾−1)||ğ‘Žğ‘š(1−𝛽)𝑛+1|||𝑧|𝑛−𝑚+1âˆ’âˆžî“ğ‘›=𝑚(𝑛+1)(1+𝛾)+𝑚(2𝛽+𝛾−1)||𝑏𝑚(1−𝛽)𝑛+1|||𝑧|𝑛−𝑚+1âˆ’âˆžî“ğ‘›=𝑚+1(𝑛+1)(1+𝛾)−𝑚(1+𝛾)||ğ‘Žğ‘š(1−𝛽)𝑛+1|||𝑧|𝑛−𝑚+1âˆ’âˆžî“ğ‘›=𝑚(𝑛+1)(1+𝛾)+𝑚(1+𝛾)𝑚||𝑏(1−𝛽)𝑛+1|||𝑧|𝑛−𝑚+1≥21âˆ’âˆžî“ğ‘›=𝑚+1(𝑛+1)(1+𝛾)−𝑚(𝛽+𝛾)||ğ‘Žğ‘š(1−𝛽)𝑛+1||âˆ’âˆžî“ğ‘›=𝑚(𝑛+1)(1+𝛾)+𝑚(𝛽+𝛾)||𝑏𝑚(1−𝛽)𝑛+1||≥0by(2.1)(2.4)
The harmonic functions 𝑓(𝑧)=𝑧𝑚+âˆžî“ğ‘›=𝑚+1𝑚(1−𝛽)𝑥(𝑛+1)(1+𝛾)−𝑚(𝛽+𝛾)𝑛𝑧𝑛+1+âˆžî“ğ‘›=𝑚𝑚(1−𝛽)(𝑛+1)(1+𝛾)+𝑚(𝛽+𝛾)𝑦𝑛𝑧𝑛+1,(2.5) where âˆ‘âˆžğ‘›=𝑚+1|𝑥𝑛∑|+âˆžğ‘›=𝑚|𝑦𝑛|=1, show that the coefficient boundary given by (2.1) is sharp. The functions of the form (2.5) are in the class 𝑅(𝑚,𝛽,𝛾) because âˆžî“ğ‘›=𝑚+1(1+𝛾)(𝑛+1)−𝑚(𝛽+𝛾)||ğ‘Žğ‘š(1−𝛽)𝑛+1||+âˆžî“ğ‘›=𝑚(1+𝛾)(𝑛+1)+𝑚(𝛽+𝛾)||𝑏𝑚(1−𝛽)𝑛+1||=âˆžî“ğ‘›=𝑚+1||𝑥𝑛||+âˆžî“ğ‘›=𝑚+1||𝑦𝑛||=1.(2.6) This completes the proof of Theorem 2.1.

In the following theorem, it is shown that the condition (2.1) is also necessary for functions 𝑓=ℎ+𝑔, where ℎ and 𝑔 are of the form (1.4).

Theorem 2.2. Let 𝑓=ℎ+𝑔 be such that ℎ and 𝑔 are given by (1.4). Then 𝑓∈𝑇(𝑚,𝛽,𝛾) if and only if âˆžî“ğ‘›=𝑚−1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)ğ‘Žğ‘š(1−𝛽)𝑛+1+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑏𝑚(1−𝛽)𝑛+1≤2.(2.7)

Proof. Since 𝑅(𝑚,𝛽,𝛾)⊂𝑇(𝑚,𝛽,𝛾), we only need to prove the “only if” part of the theorem. To this end, for functions 𝑓 of the form (1.4), we notice that the condition Re{(1+𝛾𝑒𝑖𝜃)(ğ‘§ğ‘“î…ž(𝑧))/(ğ‘§î…žğ‘“(𝑧))−𝛾𝑚𝑒𝑖𝜃}≥𝑚𝛽 is equivalent to ⎧⎪⎨⎪⎩Re1+ğ›¾ğ‘’ğ‘–ğœƒî€¸î‚€ğ‘§â„Žî…ž(𝑧)âˆ’ğ‘§ğ‘”î…žî‚î€·(𝑧)−𝑚𝛽+ğ›¾ğ‘’ğ‘–ğœƒî€¸î‚€â„Ž(𝑧)+𝑔(𝑧)ℎ(𝑧)+âŽ«âŽªâŽ¬âŽªâŽ­ğ‘”(𝑧)>0,(2.8) which implies that 𝑚Re1+𝛾𝑒𝑖𝜃−𝑚𝛽+ğ›¾ğ‘’ğ‘–ğœƒğ‘§î€¸î€»ğ‘šâˆ’âˆ‘âˆžğ‘›=𝑚+11+𝛾𝑒𝑖𝜃(𝑛+1)−𝑚𝛽+ğ›¾ğ‘’ğ‘–ğœƒğ‘Žî€¸î€»ğ‘›+1𝑧𝑛+1ğ‘§ğ‘šâˆ’âˆ‘âˆžğ‘›=𝑚+1ğ‘Žğ‘›+1𝑧𝑛+1+âˆ‘âˆžğ‘›=𝑚𝑏𝑛+1𝑧𝑛+1âˆ’âˆ‘âˆžğ‘›=𝑚1+𝛾𝑒𝑖𝜃(𝑛+1)+𝑚𝛽+𝛾𝑒𝑖𝜃𝑏𝑛+1𝑧𝑛+1ğ‘§ğ‘šâˆ’âˆ‘âˆžğ‘›=𝑚+1ğ‘Žğ‘›+1𝑧𝑛+1+âˆ‘âˆžğ‘›=𝑚𝑏𝑛+1𝑧𝑛+1𝑚∑=Re(1−𝛽)âˆ’âˆžğ‘›=𝑚+11+𝛾𝑒𝑖𝜃(𝑛+1)−𝑚𝛽+ğ›¾ğ‘’ğ‘–ğœƒğ‘Žî€¸î€»ğ‘›+1𝑧𝑛−𝑚+1∑1âˆ’âˆžğ‘›=𝑚+1ğ‘Žğ‘›+1𝑧𝑛−𝑚+1+âˆ‘âˆžğ‘›=𝑚𝑏𝑛+1𝑧𝑛−𝑚+1âˆ’âˆ‘âˆžğ‘›=𝑚1+𝛾𝑒𝑖𝜃(𝑛+1)+𝑚𝛽+𝛾𝑒𝑖𝜃𝑏𝑛+1𝑧𝑛−𝑚+1∑1âˆ’âˆžğ‘›=𝑚+1ğ‘Žğ‘›+1𝑧𝑛−𝑚+1+âˆ‘âˆžğ‘›=𝑚𝑏𝑛+1𝑧𝑛−𝑚+1>0.(2.9) Since Re(𝑒𝑖𝜃)≤|𝑒𝑖𝜃|=1, the required condition is that (2.9) is equivalent to ∑1âˆ’âˆžğ‘›=𝑚+1([](1+𝛾)(𝑛+1)−𝑚(𝛽+𝛾)/𝑚(1−𝛽))ğ‘Žğ‘›+1𝑟𝑛−𝑚+1∑1âˆ’âˆžğ‘›=𝑚+1ğ‘Žğ‘›+1𝑟𝑛−𝑚+1+âˆ‘âˆžğ‘›=𝑚𝑏𝑛+1𝑟𝑛−𝑚+1âˆ’âˆ‘âˆžğ‘›=𝑚([](1+𝛾)(𝑛+1)+𝑚(𝛽+𝛾)/𝑚(1−𝛽))𝑏𝑛+1𝑟𝑛−𝑚+1∑1âˆ’âˆžğ‘›=𝑚+1ğ‘Žğ‘›+1𝑟𝑛−𝑚+1+âˆ‘âˆžğ‘›=𝑚𝑏𝑛+1𝑟𝑛−𝑚+1≥0.(2.10) If the condition (2.7) does not hold, then the numerator in (2.10) is negative for 𝑧=𝑟 sufficiently close to 1. Hence there exists 𝑧0=𝑟0 in (0,1) for which the quotient in (2.10) is negative. This contradicts the required condition for 𝑓∈𝑇(𝑚,𝛽,𝛾), and so the proof of Theorem 2.2 is completed.

Corollary 2.3. The functions in the class 𝑇(𝑚,𝛽,𝛾)are starlike of order (𝛾+𝛽)/(1+𝛾).

Proof. The proof follows from (1.5), by putting (2.7) in the form âˆžî“ğ‘›=𝑚−1(𝑛+1)−𝑚((𝛾+𝛽)/(1+𝛾))ğ‘Žğ‘š(1−((𝛾+𝛽)/(1+𝛾)))𝑛+1+(𝑛+1)+𝑚((𝛾+𝛽)/(1+𝛾))𝑏𝑚(1−((𝛾+𝛽)/(1+𝛾)))𝑛+1≤2.(2.11)

Theorem 2.4. Let 𝑓∈𝑇(𝑚,𝛽,𝛾). Then for |𝑧|=𝑟<1, we have ||||≤𝑓(𝑧)1+𝑏𝑚+1𝑟𝑟𝑚+𝑚(1−𝛽)−𝑚(1−𝛽)+2(1+𝛾)𝑚(1+2𝛾+𝛽)+(1+𝛾)𝑏𝑚(1−𝛽)+2(1+𝛾)𝑚+1𝑟𝑚+2,||𝑓𝑚||≥(𝑧)1−𝑏𝑚+1𝑟𝑟𝑚−𝑚(1−𝛽)−𝑚(1−𝛽)+2(1+𝛾)𝑚(1+2𝛾+𝛽)+(1+𝛾)𝑏𝑚(1−𝛽)+2(1+𝛾)𝑚+1𝑟𝑚+2.(2.12)

Proof. We prove the left-hand-side inequality for |𝑓|.The proof for the right-hand-side inequality can be done by using similar arguments.
Let 𝑓∈𝑇(𝑚,𝛽,𝛾), then we have ||||=|||||𝑧𝑓(𝑧)ğ‘šâˆ’âˆžî“ğ‘›=𝑚+1ğ‘Žğ‘›+1𝑧𝑛+1+âˆžî“ğ‘›=𝑚𝑏𝑛+1𝑧𝑛+1|||||≥𝑟𝑚−𝑏𝑚+1𝑟𝑚+1âˆ’âˆžî“ğ‘›=𝑚+1î€·ğ‘Žğ‘›+1+𝑏𝑛+1𝑟𝑚+2≥𝑟𝑚−𝑏𝑚+1𝑟𝑚+1−𝑚(1−𝛽)(1+𝛾)(𝑚+2)−𝑚(𝛾+𝛽)âˆžî“ğ‘›=𝑚+1(1+𝛾)(𝑚+2)−𝑚(𝛾+𝛽)î€·ğ‘Žğ‘š(1−𝛽)𝑛+1+𝑏𝑛+1𝑟𝑛+1≥𝑟𝑚−𝑏𝑚+1𝑟𝑚+1−𝑚(1−𝛽)(1+𝛾)(𝑚+2)−𝑚(𝛾+𝛽)âˆžî“ğ‘›=𝑚+1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)ğ‘Žğ‘š(1−𝛽)𝑛+1+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑏𝑚(1−𝛽)𝑛+1𝑟𝑛+1≥1−𝑏𝑚+1𝑟𝑟𝑚−𝑚(1−𝛽)(1+𝛾)(𝑚+2)−𝑚(𝛾+𝛽)1−(1+𝛾)(𝑚+1)+𝑚(𝛾+𝛽)𝑏𝑚(1−𝛽)𝑚+1𝑟𝑚+2≥1−𝑏𝑚+1𝑟𝑟𝑚−𝑚(1−𝛽)−𝑚(1−𝛽)+2(1+𝛾)𝑚(1+2𝛾+𝛽)+(1+𝛾)𝑏𝑚(1−𝛽)+2(1+𝛾)𝑚+1𝑟𝑚+2.(2.13) This completes the proof of Theorem 2.4.

The following covering result follows from the left-side inequality in Theorem 2.4.

Corollary 2.5. Let 𝑓∈𝑇(𝑚,𝛽,𝛾),then the set 𝑤∶|𝑤|<2(1+𝛾)−𝑚(1−𝛽)+2(1+𝛾)(1+𝛾)−2𝑚(𝛾+𝛽)𝑏𝑚(1−𝛽)+2(1+𝛾)𝑚+1(2.14) is included in 𝑓(𝑈).

3. Extreme Points

Our next theorem is on the extreme points of convex hulls of the class 𝑇(𝑚,𝛽,𝛾), denoted by clco𝑇(𝑚,𝛽,𝛾).

Theorem 3.1. Let 𝑓=ℎ+𝑔 be such that ℎ and 𝑔 are given by (1.4). Then 𝑓∈clco𝑇(𝑚,𝛽,𝛾) if and only if 𝑓 can be expressed as 𝑓(𝑧)=âˆžî“ğ‘›=𝑚𝑋𝑛+1â„Žğ‘›+1(𝑧)+𝑌𝑛+1𝑔𝑛+1(,𝑧)(3.1) where â„Žğ‘š(𝑧)=𝑧𝑚,â„Žğ‘›+1(𝑧)=𝑧𝑚−𝑚(1−𝛽)(𝑧1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑛+1𝑔(𝑛=𝑚+1,𝑚+2,...),𝑛+1(𝑧)=𝑧𝑚+𝑚(1−𝛽)(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑧𝑛+1𝑋(𝑛=𝑚,𝑚+1,𝑚+2,...),𝑛+1≥0,𝑌𝑛+1≥0,âˆžî“ğ‘›=𝑚𝑋𝑛+1+𝑌𝑛+1=1.(3.2) In particular, the extreme points of the class 𝑇(𝑚,𝛽,𝛾) are {â„Žğ‘›+1} and {𝑔𝑛+1},respectively.

Proof. For functions 𝑓(𝑧) of the form (3.1), we have 𝑓(𝑧)=âˆžî“ğ‘›=𝑚𝑋𝑛+1+𝑌𝑛+1î€»ğ‘§ğ‘šâˆ’âˆžî“ğ‘›=𝑚𝑚(1−𝛽)𝑋(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑛+1𝑧𝑛+1+âˆžî“ğ‘›=𝑚𝑚(1−𝛽)𝑌(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑛+1𝑧𝑛+1.(3.3) Then âˆžî“ğ‘›=𝑚+1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑚(1−𝛽)𝑋(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑛+1+âˆžî“ğ‘›=𝑚(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑚(1−𝛽)𝑌(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑛+1=âˆžî“ğ‘›=𝑚+1𝑋𝑛+1+âˆžî“ğ‘›=𝑚𝑌𝑛+1=1−𝑋𝑚≤1,(3.4) and so 𝑓(𝑧)∈clco𝑇(𝑚,𝛽,𝛾).Conversely, suppose that 𝑓(𝑧)∈clco𝑇(𝑚,𝛽,𝛾). Set 𝑋𝑛+1=(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)ğ‘Žğ‘š(1−𝛽)𝑛+1(𝑌𝑛=𝑚+1,...),𝑛+1=(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑏𝑚(1−𝛽)𝑛+1(𝑛=𝑚,𝑚+1,...),(3.5) then note that by Theorem 2.2,0≤𝑋𝑛+1≤1(𝑛=𝑚+1,...) and 0≤𝑌𝑛+1≤1(𝑛=𝑚,𝑚+1,...).
Consequently, we obtain 𝑓(𝑧)=âˆžî“ğ‘›=𝑚𝑋𝑛+1â„Žğ‘›+1(𝑧)+𝑌𝑛+1𝑔𝑛+1(.𝑧)(3.6) Using Theorem 2.2 it is easily seen that the class 𝑇(𝑚,𝛽,𝛾)is convex and closed, and so clco𝑇(𝑚,𝛽,𝛾)=𝑇(𝑚,𝛽,𝛾).

4. Convolution Result

For harmonic functions of the form 𝑓(𝑧)=ğ‘§ğ‘šâˆ’âˆžî“ğ‘›=𝑚+1ğ‘Žğ‘›+1𝑧𝑛+1+âˆžî“ğ‘›=𝑚𝑏𝑛+1𝑧𝑛+1,(4.1)𝐺(𝑧)=ğ‘§ğ‘šâˆ’âˆžî“ğ‘›=𝑚+1𝐴𝑛+1𝑧𝑛+1+âˆžî“ğ‘›=𝑚𝐵𝑛+1𝑧𝑛+1,(4.2) we define the convolution of two harmonic functions 𝑓 and 𝐺 as (𝑓∗𝐺)(𝑧)=𝑓(𝑧)∗𝐺(𝑧)=ğ‘§ğ‘šâˆ’âˆžî“ğ‘›=𝑚+1ğ‘Žğ‘›+1𝐴𝑛+1𝑧𝑛+1+âˆžî“ğ‘›=𝑚𝑏𝑛+1𝐵𝑛+1𝑧𝑛+1.(4.3) Using this definition, we show that the class 𝑇(𝑚,𝛽,𝛾) is closed under convolution.

Theorem 4.1. For 0≤𝛽<1, let 𝑓(𝑧)∈𝑇(𝑚,𝛽,𝛾) and 𝐺(𝑧)∈𝑇(𝑚,𝛽,𝛾). Then 𝑓(𝑧)∗𝐺(𝑧)∈𝑇(𝑚,𝛽,𝛾).

Proof. Let the functions 𝑓(𝑧) defined by (4.1) be in the class𝑇(𝑚,𝛽,𝛾), and let the functions 𝐺(𝑧) defined by (4.2) be in the class 𝑇(𝑚,𝛽,𝛾). Obviously, the coefficients of 𝑓 and 𝐺 must satisfy a condition similar to the inequality (2.7). So for the coefficients of 𝑓∗𝐺we can write âˆžî“ğ‘›=𝑚−1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)ğ‘Žğ‘š(1−𝛽)𝑛+1𝐴𝑛+1+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑏𝑚(1−𝛽)𝑛+1𝐵𝑛+1â‰¤âˆžî“ğ‘›=𝑚−1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)ğ‘šğ‘Ž(1−𝛽)𝑛+1+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚𝑏(1−𝛽)𝑛+1,(4.4) where the right hand side of this inequality is bounded by 2 because 𝑓∈𝑇(𝑚,𝛽,𝛾). Then, 𝑓(𝑧)∗𝐺(𝑧)∈𝑇(𝑚,𝛽,𝛾).

Finally, we show that 𝑇(𝑚,𝛽,𝛾) is closed under convex combinations of its members.

Theorem 4.2. The class 𝑇(𝑚,𝛽,𝛾)is closed under convex combination.

Proof. For 𝑖=1,2,3,.... let 𝑓𝑖∈𝑇(𝑚,𝛽,𝛾),where the functions 𝑓𝑖 are given by 𝑓𝑖(𝑧)=ğ‘§ğ‘šâˆ’âˆžî“ğ‘›=𝑚+1ğ‘Žğ‘›+1,𝑖𝑧𝑛+1+âˆžî“ğ‘›=𝑚𝑏𝑛+1,𝑖𝑧𝑛+1î€·ğ‘Žğ‘›+1,𝑖;𝑏𝑛+1,𝑖.≥0;𝑚≥1(4.5) For âˆ‘âˆžğ‘–=1𝑡𝑖=1;0≤𝑡𝑖≤1,the convex combination of 𝑓𝑖 may be written as âˆžî“ğ‘–=1𝑡𝑖𝑓𝑖(𝑧)=ğ‘§ğ‘šâˆ’âˆžî“ğ‘›=𝑚+1îƒ©âˆžî“ğ‘–=1ğ‘¡ğ‘–ğ‘Žğ‘›+1,𝑖𝑧𝑛+1+âˆžî“ğ‘›=ğ‘šîƒ©âˆžî“ğ‘–=1𝑡𝑖𝑏𝑛+1,𝑖𝑧𝑛+1(4.6) Then by (2.7), we have âˆžî“ğ‘›=𝑚−1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑚(1−𝛽)âˆžî“ğ‘–=1ğ‘¡ğ‘–ğ‘Žğ‘›+1,𝑖+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)âˆžî“ğ‘–=1𝑡𝑖𝑏𝑛+1,𝑖=âˆžî“ğ‘–=1ğ‘¡ğ‘–îƒ¯âˆžî“ğ‘›=𝑚−1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)ğ‘Žğ‘š(1−𝛽)𝑛+1,𝑖+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑏𝑚(1−𝛽)𝑛+1,𝑖≤2âˆžî“ğ‘–=1𝑡𝑖=2.(4.7) This is the condition required by (2.7), and so âˆ‘âˆžğ‘–=1𝑡𝑖𝑓𝑖(𝑧)∈𝑇(𝑚,𝛽,𝛾).This completes the proof of Theorem 4.2.

Remark 4.3. Our results for 𝑚=1 correct the results obtained by Kharinar and More [6].

Acknowledgment

The author thanks the referees for their valuable suggestions which led to the improvement of this study.

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Copyright © 2012 R. M. El-Ashwah. 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.

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