Subclass of Multivalent Harmonic Functions with Missing Coefficients

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)|>|𝑔^(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)=𝑓^𝑧(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)=𝑧^𝑚+^∞_𝑛=_𝑚+1𝑎_𝑛+1𝑧^𝑛+1,𝑔(𝑧)=^∞_𝑛_=𝑚𝑏_𝑛+1𝑧^𝑛+1(𝑚≥1;𝑧∈𝑈)(1.1) and satisfying the following condition: Re1+𝛾𝑒^𝑖𝜙𝑧𝑓^(𝑧)𝑧^𝑓(𝑧)−𝛾𝑚𝑒^𝑖𝜙≥𝑚𝛽(𝑚≥1;0≤𝛽<1;𝛾≥0;𝜙real),(1.2) where 𝑧^=𝜕𝜕𝜃𝑧=𝑟𝑒^𝑖𝜃,𝑓^(𝑧)=𝜕𝜕𝜃𝑓𝑟𝑒^𝑖𝜃.(1.3)

We note that:(i)𝑅(𝑚,𝛽,1)=𝑅(𝑚,𝛽) with 𝑎_𝑚+1;𝑏_𝑚≠0 (see Jahangiri et al. [5]);(ii)𝑅(1,𝛽,𝛾)=𝐽_𝐻(𝛼,𝛽,𝛾) (see Kharinar and More [6]);(iii)𝑅(1,𝛽,1)=𝐺_𝐻(𝛽)with 𝑎₂;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)=𝑧^𝑚−^∞_𝑛=_𝑚+1𝑎_𝑛+1𝑧^𝑛+1,𝑔(𝑧)=^∞_𝑛_=𝑚𝑏_𝑛+1𝑧^𝑛+1𝑎_𝑛+1;𝑏_𝑛+1≥0;𝑚≥1;𝑧∈𝑈.(1.4)

We note that:(i)𝑇(𝑚,𝛽,1)=𝑇(𝑚,𝛽) with 𝑎_𝑚+1;𝑏_𝑚≠0(see Jahangiri et al. [5]);(ii)𝑇(1,𝛽,𝛾)=𝐽_𝐻(𝛼,𝛽,𝛾)(see Kharinar and More [6]);(iii)𝑇(1,𝛽,1)=𝐺_𝐻(𝛽) with 𝑎₂;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 ^∞_𝑛_=𝑚−1(𝑛𝑚+(11)−−𝛽𝑚)𝛽||𝑎^𝑛+1||+(𝑛+1)+𝑚𝛽𝑚(1−𝛽)||𝑏^𝑛+1||≤2𝑎_𝑚=1;𝑎_𝑚+1=𝑏_𝑚=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 ^∞_𝑛_=𝑚−1(1+𝛾)(𝑛𝑚+(11−)−𝛽)𝑚(𝛾+𝛽)||𝑎^𝑛+1||+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)||𝑏^𝑛+1||≤2,(2.1) where 𝑎_𝑚=1and𝑎_𝑚+1=𝑏_𝑚=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 Re⎧⎪⎨⎪⎩1+𝛾𝑒^𝑖𝜃𝑧ℎ^(𝑧)−𝑧𝑔^(𝑧)−𝑚𝛽+𝛾𝑒^𝑖𝜃ℎ(𝑧)+𝑔(𝑧)ℎ(𝑧)+𝑔(𝑧)⎫⎪⎬⎪⎭≥0,(2.2) where ℎ^(𝑧)=(𝜕/𝜕𝑧)ℎ(𝑧) and 𝑔^(𝑧)=(𝜕/𝜕𝑧)𝑔(𝑧). Substituting for ℎ,𝑔,ℎ^, and 𝑔^ in (2.2), and dividing by 𝑚(1−𝛽)𝑧^𝑚, we obtain Re(𝐴(𝑧)/𝐵(𝑧))≥0, where 𝐴(𝑧)=1+^∞_𝑛=_𝑚+1(𝑛+1)1+𝛾𝑒^𝑖𝜃−𝑚𝛽+𝛾𝑒^𝑖𝜃𝑚(1−𝛽)𝑎^𝑛+1𝑧^{𝑛−𝑚+1}−𝑧𝑧^𝑚∞_𝑛=𝑚+1(𝑛+1)1+𝛾𝑒^−𝑖𝜃+𝑚𝛽+𝛾𝑒^−𝑖𝜃𝑚(1−𝛽)𝑏^𝑛+1𝐵(𝑧)=1+_∞𝑧^{𝑛−𝑚+1},_𝑛=_𝑚+1𝑎_𝑛+1𝑧^{𝑛−𝑚+1}+𝑧𝑧^𝑚∞_𝑛=𝑚+1𝑏_𝑛+1𝑧^{𝑛−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=)+|||||2𝐵+(𝑧)_∞||−||𝐴(𝑧)−𝐵(𝑧)||_𝑛=_𝑚+1(𝑛+1)1+𝛾𝑒^𝑖𝜃−𝑚−1+2𝛽+𝛾𝑒^𝑖𝜃𝑚(1−𝛽)𝑎^𝑛+1𝑧^{𝑛−𝑚+1}−𝑧𝑧^𝑚∞_𝑛=𝑚(𝑛+1)1+𝛾𝑒^−𝑖𝜃+𝑚−1+2𝛽+𝛾𝑒^−𝑖𝜃𝑚(1−𝛽)𝑏^𝑛+1𝑧^{𝑛−𝑚+1}−|||||_∞|||||_𝑛=_𝑚+1(𝑛+1)1+𝛾𝑒^𝑖𝜃−𝑚1+𝛾𝑒^𝑖𝜃𝑚(1−𝛽)𝑎^𝑛+1𝑧^{𝑛−𝑚+1}−𝑧𝑧^∞_𝑛=𝑚(𝑛+1)1+𝛾𝑒^−𝑖𝜃+𝑚1+𝛾𝑒^−𝑖𝜃𝑚(1−𝛽)𝑏^𝑛+1𝑧^{𝑛−𝑚+1}≥2−_∞|||||_𝑛=_𝑚+1(𝑛+1)(1+𝛾)−𝑚(2𝛽+𝛾−1)𝑚(1−𝛽)||𝑎^𝑛+1|||𝑧|^{𝑛−𝑚+1}−^∞_𝑛_=𝑚(𝑛+1)(1+𝛾)+𝑚(2𝛽+𝛾−1)𝑚(1−𝛽)||𝑏^𝑛+1|||𝑧|^{𝑛−𝑚+1}−^∞_𝑛=_𝑚+1(𝑛+1)(1+𝛾)−𝑚(1+𝛾)𝑚(1−𝛽)||𝑎^𝑛+1|||𝑧|^{𝑛−𝑚+1}−^∞_𝑛_=𝑚(𝑛+1)(1+𝛾)+𝑚(1+𝛾)𝑚(1−𝛽)||𝑏^𝑛+1|||𝑧|^{𝑛−𝑚+1}≥21−^∞_𝑛=_𝑚+1(𝑛+1)(1+𝛾)−𝑚(𝛽+𝛾)𝑚(1−𝛽)||𝑎^𝑛+1−_∞||_𝑛_=𝑚(𝑛+1)(1+𝛾)+𝑚(𝛽+𝛾)𝑚(1−𝛽)||𝑏^𝑛+1≥0by(2.1)||(2.4)
The harmonic functions 𝑓(𝑧)=𝑧^𝑚+^∞_𝑛=_𝑚+1𝑚(1−𝛽)(𝑛+1)(1+𝛾)−𝑚(𝛽+𝛾)𝑥^𝑛𝑧^𝑛+1+^∞_𝑛_=𝑚𝑚(1−𝛽)(𝑛+1)(1+𝛾)+𝑚(𝛽+𝛾)𝑦_𝑛𝑧^𝑛+1,(2.5) where ∑^{∞𝑛=𝑚+1}|𝑥_𝑛|+∑^∞𝑛=𝑚|𝑦_𝑛|=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 ^∞_𝑛_=𝑚+1(1+𝛾)(𝑛𝑚+(11−)−𝛽)𝑚(𝛽+𝛾)||𝑎^𝑛+1||+^∞_𝑛_=𝑚(1+𝛾)(𝑛+1)+𝑚(𝛽+𝛾)𝑚(1−𝛽)||𝑏^𝑛+1=_∞||_𝑛=_𝑚+1||𝑥_𝑛||+^∞_𝑛=_𝑚+1||𝑦_𝑛||=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 ^∞_𝑛_=𝑚−1(1+𝛾)(𝑛𝑚+(11−)−𝛽)𝑚(𝛾+𝛽)𝑎^𝑛+1+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑏^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+𝛾𝑒^𝑖𝜃)(𝑧𝑓^(𝑧))/(𝑧^𝑓(𝑧))−𝛾𝑚𝑒^𝑖𝜃}≥𝑚𝛽 is equivalent to Re⎧⎪⎨⎪⎩1+𝛾𝑒^𝑖𝜃𝑧ℎ^(𝑧)−𝑧𝑔^(𝑧)−𝑚𝛽+𝛾𝑒^𝑖𝜃ℎ(𝑧)+𝑔(𝑧)ℎ(𝑧)+𝑔(𝑧)⎫⎪⎬⎪⎭>0,(2.8) which implies that Re𝑚1+𝛾𝑒^𝑖𝜃−𝑚𝛽+𝛾𝑒^𝑖𝜃𝑧^𝑚−∑^{∞𝑛=𝑚+1}1+𝛾𝑒^𝑖𝜃(𝑛+1)−𝑚𝛽+𝛾𝑒^𝑖𝜃𝑎_𝑛+1𝑧^𝑛+1𝑧^𝑚−∑^{∞𝑛=𝑚+1}𝑎_𝑛+1𝑧^𝑛+1+∑^∞𝑛=𝑚𝑏_𝑛+1𝑧^𝑛+1−∑^∞𝑛=𝑚1+𝛾𝑒^𝑖𝜃(𝑛+1)+𝑚𝛽+𝛾𝑒^𝑖𝜃𝑏_𝑛+1𝑧^𝑛+1=Re𝑚𝑧^𝑚(1−−∑𝛽)−∑^{∞𝑛=𝑚+1}𝑎_𝑛+1𝑧^𝑛+1_{∞𝑛=𝑚+1}+∑^∞𝑛=𝑚𝑏_𝑛+1𝑧^𝑛+11+𝛾𝑒^𝑖𝜃(𝑛+1)−𝑚𝛽+𝛾𝑒^𝑖𝜃𝑎_𝑛+1𝑧^{𝑛−𝑚+1}1−∑^{∞𝑛=𝑚+1}𝑎_𝑛+1𝑧^{𝑛−𝑚+1}+∑^∞𝑛=𝑚𝑏_𝑛+1𝑧^{𝑛−𝑚+1}−∑^∞𝑛=𝑚1+𝛾𝑒^𝑖𝜃(𝑛+1)+𝑚𝛽+𝛾𝑒^𝑖𝜃𝑏_𝑛+1𝑧^{𝑛−𝑚+1}1−∑^{∞𝑛=𝑚+1}𝑎_𝑛+1𝑧^{𝑛−𝑚+1}+∑^∞𝑛=𝑚𝑏_𝑛+1𝑧^{𝑛−𝑚+1}>0.(2.9) Since Re(𝑒^𝑖𝜃)≤|𝑒^𝑖𝜃|=1, the required condition is that (2.9) is equivalent to 1−∑^{∞𝑛=𝑚+1}([(1+𝛾)(𝑛+1)−𝑚(𝛽+𝛾)]/𝑚(1−𝛽))𝑎_𝑛+1𝑟^{𝑛−𝑚+1}1−∑^{∞𝑛=𝑚+1}𝑎_𝑛+1𝑟^{𝑛−𝑚+1}+∑^∞𝑛=𝑚𝑏_𝑛+1𝑟^{𝑛−𝑚+1}−∑^∞𝑛=𝑚([(1+𝛾)(𝑛+1)+𝑚(𝛽+𝛾)]/𝑚(1−𝛽))𝑏_𝑛+1𝑟^{𝑛−𝑚+1}1−∑^{∞𝑛=𝑚+1}𝑎_𝑛+1𝑟^{𝑛−𝑚+1}+∑^∞𝑛=𝑚𝑏_𝑛+1𝑟^{𝑛−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 ^∞_𝑛_=𝑚−1(𝑛𝑚+(11)−−(𝑚(𝛾(+(𝛾𝛽+)/𝛽()1/+(1𝛾+))𝛾)))𝑎^𝑛+1+(𝑛+1)+𝑚((𝛾+𝛽)/(1+𝛾))𝑚(1−((𝛾+𝛽)/(1+𝛾)))𝑏^𝑛+1≤2.(2.11)

Theorem 2.4. Let 𝑓∈𝑇(𝑚,𝛽,𝛾). Then for |𝑧|=𝑟<1, we have ||𝑓(𝑧)||≤1+𝑏_𝑚+1𝑟𝑟^𝑚+𝑚(1−𝛽)𝑚(1−𝛽)+2(1+𝛾)−𝑚𝑚(1(+1−2𝛾𝛽+)+𝛽)2+(1(+1+𝛾)𝛾)𝑏^𝑚+1𝑟^𝑚+2||𝑓_𝑚,(𝑧)||≥1−𝑏_𝑚+1𝑟𝑟^𝑚−𝑚(1−𝛽)𝑚(1−𝛽)+2(1+𝛾)−𝑚𝑚(1(+1−2𝛾𝛽+)+𝛽)2+(1(+1+𝛾)𝛾)𝑏^𝑚+1𝑟^𝑚+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 ||𝑓(𝑧)||=|||||𝑧^𝑚−^∞_𝑛=_𝑚+1𝑎_𝑛+1𝑧^𝑛+1+^∞_𝑛_=𝑚𝑏_𝑛+1𝑧^𝑛+1≥𝑟_𝑚|||||−𝑏_𝑚+1𝑟^𝑚+1−^∞_𝑛=_𝑚+1𝑎_𝑛+1+𝑏_𝑛+1𝑟^𝑚+2≥𝑟^𝑚−𝑏_𝑚+1𝑟^𝑚+1−𝑚(1−𝛽)(1+𝛾)(𝑚+2)−𝑚(𝛾+𝛽)^∞_𝑛=𝑚+1(1+𝛾)(𝑚+2)−𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑎^𝑛+1+𝑏_𝑛+1𝑟^𝑛+1≥𝑟^𝑚−𝑏_𝑚+1𝑟^𝑚+1−𝑚(1−𝛽)(1+𝛾)(𝑚+2)−𝑚(𝛾+𝛽)^∞_𝑛=𝑚+1(1+𝛾)(𝑛𝑚+(11)−−𝛽𝑚)(𝛾+𝛽)𝑎^𝑛+1+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑏^𝑛+1𝑟^𝑛+1≥1−𝑏_𝑚+1𝑟𝑟^𝑚−𝑚(1−𝛽)(1+𝛾)(𝑚+2)−𝑚(𝛾+𝛽)1−(1+𝛾)(𝑚𝑚+(11−)+𝛽)𝑚(𝛾+𝛽)𝑏^𝑚+1𝑟^𝑚+2≥1−𝑏_𝑚+1𝑟𝑟^𝑚−𝑚(1−𝛽)𝑚(1−𝛽)+2(1+𝛾)−𝑚𝑚(1(+1−2𝛾𝛽+)+𝛽)2+(1(+1+𝛾)𝛾)𝑏^𝑚+1𝑟^𝑚+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|<𝑚(1−2𝛽()1++2𝛾()1+𝛾)−𝑚(1(+1−𝛾)𝛽−)+2𝑚2((𝛾1++𝛽𝛾))𝑏^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)=^∞_𝑛_=𝑚𝑋_𝑛+1ℎ_𝑛+1(𝑧)+𝑌_𝑛+1𝑔_𝑛+1(𝑧),(3.1) where ℎ_𝑛+1ℎ_𝑚(𝑧)=𝑧^𝑚,(𝑧)=𝑧^𝑚−𝑚(1−𝛽)(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑧^𝑛+1𝑔(𝑛=𝑚+1,𝑚+2,...),_𝑛+1(𝑧)=𝑧^𝑚+𝑚(1−𝛽)(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑋𝑧^𝑛+1(𝑛=𝑚,𝑚+1,𝑚+2,...),_𝑛+1≥0,𝑌_𝑛+1≥0,^∞_𝑛_=𝑚𝑋_𝑛+1+𝑌_𝑛+1=1.(3.2) In particular, the extreme points of the class 𝑇(𝑚,𝛽,𝛾) are {ℎ_𝑛+1} and {𝑔_𝑛+1},respectively.

Proof. For functions 𝑓(𝑧) of the form (3.1), we have 𝑓(𝑧)=^∞_𝑛_=𝑚𝑋_𝑛+1+𝑌_𝑛+1𝑧^𝑚−^∞_𝑛_=𝑚𝑚(1−𝛽)(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑋^𝑛+1𝑧^𝑛+1+^∞_𝑛_=𝑚𝑚(1−𝛽)(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑌^𝑛+1𝑧^𝑛+1.(3.3) Then ^∞_𝑛_=𝑚+1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑚(1−𝛽)(1+𝛾)(𝑛𝑚+(11−)−𝛽)𝑚(𝛾+𝛽)𝑋^𝑛+1+^∞_𝑛_=𝑚(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)(1+𝛾)(𝑛𝑚+(11−)+𝛽)𝑚(𝛾+𝛽)𝑌^𝑛+1=^∞_𝑛=_𝑚+1𝑋_𝑛+1+^∞_𝑛_=𝑚𝑌_𝑛+1=1−𝑋_𝑚≤1,(3.4) and so 𝑓(𝑧)∈clco𝑇(𝑚,𝛽,𝛾).Conversely, suppose that 𝑓(𝑧)∈clco𝑇(𝑚,𝛽,𝛾). Set 𝑋_𝑛+1=(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑎^𝑛+1𝑌(𝑛=𝑚+1,...),_𝑛+1=(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑏^𝑛+1(𝑛=𝑚,𝑚+1,...),(3.5) then note that by Theorem 2.2,0≤𝑋_𝑛+1≤1(𝑛=𝑚+1,...) and 0≤𝑌_𝑛+1≤1(𝑛=𝑚,𝑚+1,...).
Consequently, we obtain 𝑓(𝑧)=^∞_𝑛_=𝑚𝑋_𝑛+1ℎ_𝑛+1(𝑧)+𝑌_𝑛+1𝑔_𝑛+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)=𝑧^𝑚−^∞_𝑛=_𝑚+1𝑎_𝑛+1𝑧^𝑛+1+^∞_𝑛_=𝑚𝐺(𝑧)=𝑧_𝑚𝑏_𝑛+1𝑧^𝑛+1,(4.1)−^∞_𝑛=_𝑚+1𝐴_𝑛+1𝑧^𝑛+1+^∞_𝑛_=𝑚𝐵_𝑛+1𝑧^𝑛+1,(4.2) we define the convolution of two harmonic functions 𝑓 and 𝐺 as (𝑓∗𝐺)(𝑧)==𝑧𝑓_𝑚(𝑧)∗𝐺(𝑧)−^∞_𝑛=_𝑚+1𝑎_𝑛+1𝐴_𝑛+1𝑧^𝑛+1+^∞_𝑛_=𝑚𝑏_𝑛+1𝐵_𝑛+1𝑧^𝑛+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 ^∞_𝑛_=𝑚−1(1+𝛾)(𝑛+1)−𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑎^𝑛+1𝐴_𝑛+1+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑏^𝑛+1𝐵_𝑛+1≤^∞_𝑛=_𝑚−1(1+𝛾)(𝑛𝑚+(11−)−𝛽)𝑚(𝛾+𝛽)𝑎^𝑛+1+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)𝑏^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 𝑓_𝑖∈𝑇(𝑚,𝛽,𝛾),where the functions 𝑓_𝑖 are given by 𝑓_𝑖(𝑧)=𝑧^𝑚−^∞_𝑛=_𝑚+1𝑎_𝑛+1,𝑖𝑧^𝑛+1+^∞_𝑛_=𝑚𝑏_𝑛+1,𝑖𝑧^𝑛+1𝑎_𝑛+1,𝑖;𝑏_𝑛+1,𝑖≥0;𝑚≥1.(4.5) For ∑^∞𝑖=1𝑡_𝑖=1;0≤𝑡_𝑖≤1,the convex combination of 𝑓_𝑖 may be written as ^∞_𝑖=1𝑡_𝑖𝑓_𝑖(𝑧)=𝑧^𝑚−^∞_𝑛=_𝑚+1^∞_𝑖=1𝑡_𝑖𝑎_𝑛+1,𝑖𝑧^𝑛+1+^∞_𝑛_=𝑚^∞_𝑖=1𝑡_𝑖𝑏_𝑛+1,𝑖𝑧^𝑛+1(4.6) Then by (2.7), we have ^∞_𝑛_=𝑚−1(1+𝛾)(𝑛𝑚+(11−)−𝛽)𝑚(𝛾+𝛽)^∞_𝑖=1𝑡_𝑖𝑎_𝑛+1,𝑖+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)𝑚(1−𝛽)^∞_𝑖=1𝑡_𝑖𝑏_𝑛+1,𝑖=_∞_𝑖=1𝑡_𝑖^∞_{𝑛=𝑚−1}(1+𝛾)(𝑛𝑚+(11−)−𝛽)𝑚(𝛾+𝛽)𝑎^𝑛+1,𝑖+(1+𝛾)(𝑛+1)+𝑚(𝛾+𝛽)≤2_∞𝑚(1−𝛽)𝑏^𝑛+1,𝑖_𝑖=1𝑡_𝑖=2.(4.7) This is the condition required by (2.7), and so ∑^∞𝑖=1𝑡_𝑖𝑓_𝑖(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.