A Convolution Approach on Partial Sums of Certain Harmonic Univalent Functions

Abstract

The purpose of the present paper is to establish some new results giving the sharp bounds of the real parts of ratios of harmonic univalent functions to their sequences of partial sums by using convolution. Relevant connections of the results presented here with various known results are briefly indicated.

1. Introduction

A continuous complex-valued function 𝑓=𝑢+𝑖𝑣 is said to be harmonic in a simply connected domain 𝐷 if both 𝑢 and 𝑣 are real harmonic in 𝐷. In any simply-connected domain we can write 𝑓=ℎ+𝑔, where ℎ and 𝑔 are analytic in 𝐷. We call ℎ the analytic part and 𝑔 the co-analytic part of 𝑓. A necessary and sufficient condition for 𝑓 to be locally univalent and sense-preserving in 𝐷 is that |ℎ^′(𝑧)|>|𝑔^′(𝑧)|,𝑧∈𝐷, see [1]. For more basic results on harmonic functions one may refer to the following standard text book by Duren [2]. See also Ahuja [3] and Ponnusamy and Rasila ([4, 5]).

Denote by 𝑆_𝐻 the class of functions 𝑓=ℎ+𝑔 which are harmonic univalent and sense-preserving in the open unit disk 𝑈={𝑧∶|𝑧|<1} for which 𝑓(0)=𝑓_𝑧(0)−1=0. Then for 𝑓=ℎ+𝑔∈𝑆_𝐻 we may express the analytic functions ℎ and 𝑔 as ℎ(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘,𝑔(𝑧)=^∞_𝑘=1𝑏_𝑘𝑧^𝑘,𝑏₁<1. (1.1)

Note that 𝑆_𝐻 reduces to the class 𝑆 of normalized analytic univalent functions, if the coanalytic part of its member is zero, that is, 𝑔≡0, and for this class 𝑓(𝑧) may be expressed as 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘. (1.2)

Let 𝜙(𝑧)∈𝑆_𝐻 be a fixed function of the form 𝜙(𝑧)=𝑧+^∞_𝑘=2𝑐_𝑘𝑧^𝑘+^∞_𝑘=1𝑑_𝑘𝑧^𝑘,𝑑_𝑘≥𝑐_𝑘≥𝑐₂>0;𝑘≥2,𝑑₁<1. (1.3)

Now, we introduce a class 𝑆_𝐻(𝑐_𝑘,𝑑_𝑘,𝛿) consisting of functions of the form (1.1) which satisfies the inequality ^∞_𝑘=2𝑐_𝑘𝑎_𝑘+^∞_𝑘=1𝑑_𝑘𝑏_𝑘≤𝛿,where𝛿>0, (1.4) and we note that if 𝑑_𝑘=0, then the class 𝑆_𝐻(𝑐_𝑘,𝑑_𝑘,𝛿) reduces to the class 𝑆_𝜙(𝑐_𝑘,𝛿) which was introduced by Frasin [6]. In this case the condition (1.4) reduces to ^∞_𝑘=2𝑐_𝑘𝑎_𝑘≤𝛿,where𝛿>0. (1.5)

It is easy to see that various subclasses of 𝑆_𝐻 consisting of functions 𝑓(𝑧) of the form (1.1) can be represented as 𝑆_𝐻(𝑐_𝑘,𝑑_𝑘,𝛿) for suitable choices of 𝑐_𝑘,𝑑_𝑘, and 𝛿 studied earlier by various researchers. For example: (1)𝑆_𝐻(𝑘,𝑘,1)≡𝑆^∗_𝐻 and 𝑆_𝐻(𝑘²,𝑘²,1)≡𝐾_𝐻 studied by Silverman [7]; Silverman and Silvia [8]. (2)𝑆_𝐻(𝑘−𝛼,𝑘+𝛼,1−𝛼)≡𝑆^∗_𝐻(𝛼) studied by Jahangiri [9]. (3)𝑆_𝐻((𝑘−𝛼)(𝜙(𝑘,𝜆)),(𝑘+𝛼)(𝜙(𝑘,𝜆)),1−𝛼)≡𝑆^∗_𝐻,𝜆(𝛼) studied by Dixit and Porwal [10]. (4)𝑆_𝐻(𝑘^𝑚−𝛼𝑘^𝑛,𝑘^𝑚−𝛼𝑘^𝑛,1−𝛼)≡𝐻𝑆(𝑚,𝑛,𝛼) studied by Dixit and Porwal [11]. (5)𝑆_𝐻(𝑘,𝑘,𝛽−1)≡𝐻𝑃(𝛽) studied by Dixit and Porwal [12]. (6)𝑆_𝐻(𝜆_𝑘(1−𝛼𝜆)−𝛼(1−𝜆),𝜇_𝑘(1−𝛼𝜆)+𝛼(1−𝜆),1−𝛼)≡𝑆_𝐻(Φ,Ψ,𝛼,𝜆) studied by Dixit and Porwal [13]. (7)𝑆_𝐻(𝜆_𝑘−𝛼,𝜇_𝑘+𝛼,1−𝛼)≡𝑆_𝐻(Φ,Ψ,𝛼) studied by Frasin [14]. (8)𝑆_𝐻(𝑘(1−𝛼𝜆)−𝛼(1−𝜆),𝑘(1−𝛼𝜆)+𝛼(1−𝜆),1−𝛼)≡𝑆^∗_𝐻(𝛼,𝜆) studied by Öztürk et al. [15]. (9)𝑆_𝐻((𝑘(𝛽+1)−𝑡(𝛽+𝛾))Γ(𝛼₁,𝑘),(𝑘(𝛽+1)+𝑡(𝛽+𝛾))Γ(𝛼₁,𝑘)),(𝑘−1)!(1−𝛾))≡𝐺_𝐻(𝛼₁,𝛽,𝛾,𝑡) studied by Porwal et al. [16]. (10)𝑆_𝐻(2𝑘−1−𝛼,2𝑘+1+𝛼,1−𝛼)≡𝐺_𝐻(𝛼) studied by Rosy et al. [17]. In 1985, Silvia [18] studied the partial sums of convex functions of order 𝛼. Later on, Silverman [19], Afaf et al. [20], Dixit and Porwal [21], Frasin ([6, 22]), Murugusundaramoorthy et al. [23], Owa et al. [24], Porwal and Dixit [25], Raina and Bansal [26] and Rosy et al. [27] studied and generalized the results on partial sums for various classes of analytic functions. Very recently, Porwal [28], Porwal and Dixit [29] studied analogues interesting results on the partial sums of certain harmonic univalent functions. In this work, we extend all these results.

Now, we let the sequences of partial sums of function of the form (1.1) with 𝑏₁=0 be 𝑓_𝑚(𝑧)=𝑧+^𝑚_𝑘=2𝑎_𝑘𝑧^𝑘+^∞_𝑘=2𝑏_𝑘𝑧^𝑘,𝑓_𝑛(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘+^𝑛_𝑘=2𝑏_𝑘𝑧^𝑘,𝑓_𝑚,𝑛(𝑧)=𝑧+^𝑚_𝑘=2𝑎_𝑘𝑧^𝑘+^𝑛_𝑘=2𝑏_𝑘𝑧^𝑘, (1.6) when the coefficients of 𝑓 are sufficiently small to satisfy the condition (1.4).

In the present paper, we determine sharp lower bounds for Re{(𝑓(𝑧)∗𝜓(𝑧))/(𝑓_𝑚(𝑧)∗𝜓(𝑧))}, Re{(𝑓_𝑚(𝑧)∗𝜓(𝑧))/(𝑓(𝑧)∗𝜓(𝑧))}, Re{(𝑓(𝑧)∗𝜓(𝑧))/(𝑓_𝑛(𝑧)∗𝜓(𝑧))}, Re{(𝑓_𝑛(𝑧)∗𝜓(𝑧))/(𝑓(𝑧)∗𝜓(𝑧))}, Re{(𝑓(𝑧)∗𝜓(𝑧))/(𝑓_𝑚,𝑛(𝑧)∗𝜓(𝑧))}, and Re{(𝑓_𝑚,𝑛(𝑧)∗𝜓(𝑧))/(𝑓(𝑧)∗𝜓(𝑧))} where 𝑓_𝑚(𝑧), 𝑓_𝑛(𝑧) and 𝑓_𝑚,𝑛(𝑧) are defined above and 𝜓(𝑧)=𝑧+∑^∞_𝑘=2𝜆_𝑘𝑧^𝑘+∑^∞_𝑘=2𝜇_𝑘𝑧^𝑘, (𝜆_𝑘≥0,𝜇_𝑘≥0) is a harmonic function and the operator “∗” stands for the Hadamard product or convolution of two power series, which is defined for two functions 𝑓(𝑧) and 𝑔(𝑧) are of the form 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘+^∞_𝑘=2𝑏_𝑘𝑧^𝑘,𝑔(𝑧)=𝑧+^∞_𝑘=2𝑐_𝑘𝑧^𝑘+^∞_𝑘=2𝑑_𝑘𝑧^𝑘, (1.7) as 𝑓∗𝑔(𝑧)=𝑓(𝑧)∗𝑔(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑐_𝑘𝑧^𝑘+^∞_𝑘=2𝑏_𝑘𝑑_𝑘𝑧^𝑘. (1.8) It is worthy to note that this study not only gives as a particular case, the results of Porwal [28], Porwal and Dixit [29], but also give rise to several new results.

2. Main Results

In our first theorem, we determine sharp lower bounds for Re{(𝑓(𝑧)∗𝜓(𝑧))/(𝑓_𝑚(𝑧)∗𝜓(𝑧))}.

Theorem 2.1. If 𝑓 of the form (1.1) with 𝑏₁=0, satisfies the condition (1.4), then 𝑓(𝑧)(𝑧)Re∗𝜓𝑓_𝑚(𝑧)(𝑧)≥𝑐∗𝜓^𝑚+1−𝜆_𝑚+1𝛿𝑐_𝑚+1,(𝑧∈𝑈), (2.1) where 𝑐_𝑘≥𝜆_𝑘𝜆𝛿if𝑘=2,3,…,𝑚,_𝑘𝑐_𝑚+1𝜆_𝑚+1if𝑘=𝑚+1,𝑚+2…. (2.2)
The result (2.1) is sharp with the function given by 𝑓(𝑧)𝛿=𝑧+𝑐_𝑚+1𝑧^𝑚+1, (2.3) where 0<𝛿≤𝑐_𝑚+1/𝜆_𝑚+1.

Proof. To obtain sharp lower bound given by (2.1), let us put 1+𝜔(𝑧)(𝑧)=𝑐1−𝜔_𝑚+1𝜆_𝑚+1𝛿𝑓𝑟𝑒^𝑖𝜃∗𝜓𝑟𝑒^𝑖𝜃𝑓_𝑚𝑟𝑒^𝑖𝜃∗𝜓𝑟𝑒^𝑖𝜃−𝑐_𝑚+1−𝜆_𝑚+1𝛿𝑐_𝑚+1=1+∑^𝑚_𝑘=2𝜆_𝑘𝑎_𝑘𝑟^𝑘−1𝑒^{𝑖(𝑘−1)𝜃}+∑^∞_𝑘=2𝜇_𝑘𝑏_𝑘𝑟^𝑘−1𝑒^{−𝑖(𝑘+1)𝜃}+𝑐_𝑚+1/𝜆_𝑚+1𝛿∑^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘𝑟^𝑘−1𝑒^{𝑖(𝑘−1)𝜃}1+∑^𝑚_𝑘=2𝜆_𝑘𝑎_𝑘𝑟^𝑘−1𝑒^{𝑖(𝑘−1)𝜃}+∑^∞_𝑘=2𝜇_𝑘𝑏_𝑘𝑟^𝑘−1𝑒^{−𝑖(𝑘+1)𝜃}. (2.4) So that 𝜔(𝑧)=ℛ∑^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘𝑟^𝑘−1𝑒^{𝑖(𝑘−1)𝜃}2+2∑^𝑚_𝑘=2𝜆_𝑘𝑎_𝑘𝑟^𝑘−1𝑒^{𝑖(𝑘−1)𝜃}+∑^∞_𝑘=2𝜇_𝑘𝑏_𝑘𝑟^𝑘−1𝑒^{−𝑖(𝑘+1)𝜃}+ℛ∑^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘𝑟^𝑘−1𝑒^{𝑖(𝑘−1)𝜃}, (2.5) where ℛ denotes (𝑐_𝑚+1/𝜆_𝑚+1𝛿).
Then |𝜔(𝑧≤𝑐)|^𝑚+1/𝜆_𝑚+1𝛿∑^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘2−2∑^𝑚_𝑘=2𝜆_𝑘𝑎_𝑘+∑^∞_𝑘=2𝜇_𝑘𝑏_𝑘−𝑐_𝑚+1/𝜆_𝑚+1𝛿∑^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘. (2.6) This last expression is bounded above by 1, if and only if ^𝑚_𝑘=2𝜆_𝑘𝑎_𝑘+^∞_𝑘=2𝜇_𝑘𝑏_𝑘+𝑐_𝑚+1𝜆_𝑚+1𝛿^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘≤1. (2.7) It suffices to show that L.H.S. of (2.7) is bounded above by  ∑^∞_𝑘=2(𝑐_𝑘/𝛿)|𝑎_𝑘|+∑^∞_𝑘=2(𝑑_𝑘/𝛿)|𝑏_𝑘|, which is equivalent to ^𝑚_𝑘=2𝑐_𝑘−𝜆_𝑘𝛿𝛿𝑎_𝑘+^∞_𝑘=2𝑑_𝑘−𝜇_𝑘𝛿𝛿𝑏_𝑘+^∞_𝑘=𝑚+1𝑐_𝑘𝜆_𝑚+1−𝑐_𝑚+1𝜆_𝑘𝜆_𝑚+1𝛿𝑎_𝑘≥0. (2.8)
To see that 𝑓(𝑧)=𝑧+(𝛿/𝑐_𝑚+1)𝑧^𝑚+1 gives the sharp result, we observe that for 𝑧=𝑟𝑒^{𝑖𝜋/𝑚} that 𝑓(𝑧)∗𝜓(𝑧)𝑓_𝑚(𝑧)(𝑧)𝜆∗𝜓=1+^𝑚+1𝛿𝑐_𝑚+1𝑧^𝑚𝜆→1−^𝑚+1𝛿𝑐_𝑚+1=𝑐_𝑚+1−𝜆_𝑚+1𝛿𝑐_𝑚+1, (2.9) when 𝑟→1⁻.

We next determine bounds for Re{(𝑓_𝑚(𝑧)∗𝜓(𝑧))/(𝑓(𝑧)∗𝜓(𝑧))}.

Theorem 2.2. If 𝑓 of the form (1.1) with 𝑏₁=0, satisfies the condition (1.4), then 𝑓Re^𝑚(𝑧)∗𝜓(𝑧)𝑓(𝑧)(𝑧)≥𝑐∗𝜓𝑐_𝑚+1+𝜆^𝑚+1_𝑚+1𝛿,(𝑧∈𝑈), (2.10) where 𝑐_𝑘≥𝜆_𝑘𝜆𝛿if𝑘=2,3,…,𝑚_𝑘𝑐_𝑚+1𝜆_𝑚+1if𝑘=𝑚+1,𝑚+2…. (2.11) The result (2.10) is sharp with the function given by (2.3).

Proof. To prove Theorem 2.2, we may write 1+𝜔(𝑧)(𝑧)=𝑐1−𝜔_𝑚+1+𝜆_𝑚+1𝛿𝜆_𝑚+1𝛿𝑓_𝑚(𝑧)∗𝜓(𝑧)𝑓(𝑧)(𝑧)−𝑐∗𝜓𝑐_𝑚+1+𝜆^𝑚+1_𝑚+1𝛿=1+∑^𝑚_𝑘=2𝜆_𝑘𝑎_𝑘𝑟^𝑘−1𝑒^{𝑖(𝑘−1)𝜃}+∑^∞_𝑘=2𝜇_𝑘𝑏_𝑘𝑟^𝑘−1𝑒^{−𝑖(𝑘+1)𝜃}−𝑐_𝑚+1/𝜆_𝑚+1𝛿∑^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘𝑟^𝑘−1𝑒^{𝑖(𝑘−1)𝜃}1+∑^∞_𝑘=2𝜆_𝑘𝑎_𝑘𝑟^𝑘−1𝑒^{𝑖(𝑘−1)𝜃}+∑^∞_𝑘=2𝜇_𝑘𝑏_𝑘𝑟^𝑘−1𝑒^{−𝑖(𝑘+1)𝜃}, (2.12) where |𝜔(𝑧≤𝑐)|^𝑚+1+𝜆_𝑚+1𝛿/𝜆_𝑚+1𝛿∑^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘2−2∑^𝑚_𝑘=2𝜆_𝑘𝑎_𝑘+∑^∞_𝑘=2𝜇_𝑘𝑏_𝑘−𝑐_𝑚+1−𝜆_𝑚+1𝛿/𝜆_𝑚+1𝛿∑^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘≤1. (2.13) This last inequality is equivalent to ^𝑚_𝑘=2𝜆_𝑘𝑎_𝑘+^∞_𝑘=2𝜇_𝑘𝑏_𝑘+𝑐_𝑚+1𝜆_𝑚+1𝛿^∞_𝑘=𝑚+1𝜆_𝑘𝑎_𝑘≤1. (2.14) Since the L.H.S. of (2.14) is bounded above by ∑^∞_𝑘=2(𝑐_𝑘/𝛿)|𝑎_𝑘|+∑^∞_𝑘=2(𝑑_𝑘/𝛿)|𝑏_𝑘|, the proof is evidently complete.

Adopting the same procedure as in Theorems 2.1 and 2.2 and performing simple calculations, we can obtain the sharp lower bounds for the real parts of the following ratios: 𝑓(𝑧)(𝑧Re∗𝜓)𝑓_𝑛(𝑧)(𝑧)𝑓∗𝜓,Re^𝑛(𝑧)∗𝜓(𝑧)𝑓(𝑧)(𝑧)𝑓(𝑧)(𝑧∗𝜓,Re∗𝜓)𝑓_𝑚,𝑛(𝑧)(𝑧),𝑓∗𝜓Re^𝑚,𝑛(𝑧)∗𝜓(𝑧)𝑓(𝑧)(𝑧).∗𝜓 (2.15)

The results corresponding to real parts of these ratios are contained in the following Theorems 2.3, 2.4, 2.5, and 2.6.

Theorem 2.3. If 𝑓 of the form (1.1) with 𝑏₁=0 satisfies the condition (1.4), then 𝑓(𝑧)(𝑧)Re∗𝜓𝑓_𝑛(𝑧)(𝑧)≥𝑑∗𝜓^𝑛+1−𝜇_𝑛+1𝛿𝑑_𝑛+1,(𝑧∈𝑈), (2.16) where 𝑑_𝑘≥𝜇_𝑘𝜇𝛿if𝑘=2,3,…,𝑛,_𝑘𝑑_𝑛+1𝜇_𝑛+1if𝑘=𝑛+1,𝑛+2…. (2.17)
The result (2.16) is sharp with the function 𝑓(𝑧)𝛿=𝑧+𝑑_𝑛+1𝑧^𝑛+1. (2.18)

Theorem 2.4. If 𝑓 of the form (1.1) with 𝑏₁=0, satisfies the condition (1.4), then 𝑓Re^𝑛(𝑧)∗𝜓(𝑧)𝑓(𝑧)(𝑧)≥𝑑∗𝜓𝑑_𝑛+1+𝜇^𝑛+1_𝑛+1𝛿,(𝑧∈𝑈), (2.19) where 𝑑_𝑘≥𝜇_𝑘𝜇𝛿if𝑘=2,3,…,𝑛,_𝑘𝑑_𝑛+1𝜇_𝑛+1if𝑘=𝑛+1,𝑛+2…. (2.20)
The result (2.19) is sharp with the function given by (2.18).

Theorem 2.5. If 𝑓 of the form (1.1) with 𝑏₁=0, satisfies the condition (1.4), then(i)𝑓(𝑧)(𝑧)Re∗𝜓𝑓_𝑚,𝑛(𝑧)(𝑧)≥𝑐∗𝜓^𝑚+1−𝜆_𝑚+1𝛿𝑐_𝑚+1,(𝑧∈𝑈), (2.21) where 𝑐_𝑘≥𝜆_𝑘𝜆𝛿if𝑘=2,3,…,𝑚,_𝑘𝑐_𝑚+1𝜆_𝑚+1𝑑if𝑘=𝑚+1,𝑚+2…,_𝑘≥𝜆_𝑘𝜆𝛿if𝑘=2,3,…,𝑚,_𝑘𝑐_𝑚+1𝜆_𝑚+1if𝑘=𝑚+1,𝑚+2… (2.22)(ii)𝑓(𝑧)(𝑧)Re∗𝜓𝑓_𝑚,𝑛(𝑧)(𝑧)≥𝑑∗𝜓^𝑛+1−𝜇_𝑛+1𝛿𝑑_𝑛+1,(𝑧∈𝑈), (2.23) where 𝑐_𝑘≥𝜇_𝑘𝜇𝛿if𝑘=2,3,…,𝑛,_𝑘𝑑_𝑛+1𝜇_𝑛+1𝑑if𝑘=𝑛+1,𝑛+2…,_𝑘≥𝜇_𝑘𝜇𝛿if𝑘=2,3,…,𝑛,_𝑘𝑑_𝑛+1𝜇_𝑛+1if𝑘=𝑛+1,𝑛+2…. (2.24) The results (2.21) and (2.23) are sharp with the functions given by (2.3) and (2.18), respectively.

Theorem 2.6. If 𝑓 of the form (1.1) with 𝑏₁=0, satisfies condition (1.4), then(i)𝑓Re^𝑚,𝑛(𝑧)∗𝜓(𝑧)𝑓(𝑧)(𝑧)≥𝑐∗𝜓𝑐_𝑚+1+𝜆^𝑚+1_𝑚+1𝛿,(𝑧∈𝑈), (2.25) where 𝑐_𝑘≥𝜆_𝑘𝜆𝛿if𝑘=2,3,…,𝑚,_𝑘𝑐_𝑚+1𝜆_𝑚+1𝑑if𝑘=𝑚+1,𝑚+2…,_𝑘≥𝜆_𝑘𝜆𝛿if𝑘=2,3,…,𝑚,_𝑘𝑐_𝑚+1𝜆_𝑚+1if𝑘=𝑚+1,𝑚+2…. (2.26)(ii)𝑓Re^𝑚,𝑛(𝑧)∗𝜓(𝑧)𝑓(𝑧)(𝑧)≥𝑑∗𝜓𝑑_𝑛+1+𝜇^𝑛+1_𝑛+1𝛿,(𝑧∈𝑈), (2.27) where 𝑐_𝑘≥𝜇_𝑘𝜇𝛿if𝑘=2,3,…,𝑚,_𝑘𝑑_𝑛+1𝜇_𝑛+1𝑑if𝑘=𝑚+1,𝑚+2…,_𝑘≥𝜇_𝑘𝜇𝛿if𝑘=2,3,…,𝑛,_𝑘𝑑_𝑛+1𝜇_𝑛+1if𝑘=𝑛+1,𝑛+2…. (2.28) The results (2.25) and (2.27) are sharp with the functions given by (2.3) and (2.18) respectively.

3. Some Consequences and Concluding Remarks

In this section, we specifically point out the relevances of some of our main results with those results which have appeared recently in literature.

If we put 𝜓(𝑧)=(𝑧/(1−𝑧))+((𝑧/(1−𝑧))−𝑧) and 𝜓(𝑧)=(𝑧/(1−𝑧)²)+((𝑧/(1−𝑧)²)−𝑧) in Theorems 2.1–2.6, then we obtain the corresponding results of Porwal [28].

Next, if we put 𝜓(𝑧)=(𝑧/(1−𝑧))+((𝑧/(1−𝑧))−𝑧), 𝜓(𝑧)=(𝑧/(1−𝑧)²)+((𝑧/(1−𝑧)²)−𝑧), 𝑐_𝑘=𝑘−𝛼,𝑑_𝑘=𝑘+𝛼, and 𝛿=1−𝛼 in Theorems 2.1–2.6, then we obtain the corresponding results of Porwal and Dixit [29].

Again, if we put 𝑔=0 in Theorems 2.1 and 2.2, then we obtain the corresponding results of Dixit and Porwal [21].

Lastly, if we put 𝑔≡0, 𝜓(𝑧)=𝑧/(1−𝑧), and 𝜓(𝑧)=𝑧/(1−𝑧)² Theorems 2.1 and 2.2, then we obtain the result of Frasin [6].

We mention below some corollaries giving sharp bounds of the real parts on the ratio of univalent functions to its sequences of partial sums.

By putting 𝜓(𝑧)=𝑧/(1−𝑧) in Theorem 2.1 for the function 𝑓 of the form (1.2) with 𝑐_𝑘=𝑘−𝛼 and 𝛿=1−𝛼, then we obtain the following result of Silverman [19], Theorem 1.

Corollary 3.1. If 𝑓 of the form (1.2) satisfies the condition (1.5) with 𝑐_𝑘=𝑘−𝛼 and 𝛿=1−𝛼, then 𝑓(𝑧)Re𝑓_𝑚(𝑧)≥𝑚𝑚+1−𝛼,𝑧∈𝑈. (3.1) The result is sharp for every 𝑚, with the extremal function given by 𝑓(𝑧)=𝑧+1−𝛼𝑧𝑚+1−𝛼^𝑚+1. (3.2)

On the other hand, if we put 𝜓(𝑧)=𝑧/(1−𝑧)² in Theorem 2.1 for the function 𝑓 of the form (1.2) with 𝑐_𝑘=𝑘−𝛼 and 𝛿=1−𝛼, then we obtain the following result of Silverman, Theorem 4(i) [19].

Corollary 3.2. If 𝑓 of the form (1.2) satisfies the condition (1.5) with 𝑐_𝑘=𝑘−𝛼, then for 𝑧∈𝑈𝑓Re^′(𝑧)𝑓^′_𝑚(𝑧)≥𝛼𝑚.𝑚+1−𝛼 (3.3) The result is sharp for every 𝑚, with the extremal function given by (3.2).

Also, if we put 𝜓(𝑧)=𝑧/(1−𝑧) in Theorem 2.1 for the function 𝑓 of the form (1.2) belonging to the class 𝑆_𝜙(𝑐_𝑘,𝛿), then we obtain the following result of Frasin [6].

Corollary 3.3. If ∈𝑆_𝜙(𝑐_𝑘,𝛿), then 𝑓(𝑧)Re𝑓_𝑚(𝑧)≥𝑐_𝑚+1−𝛿𝑐_𝑚+1,(𝑧∈𝑈), (3.4) where 𝑐_𝑘≥𝑐𝛿if𝑘=2,3,…,𝑚,_𝑚+1if𝑘=𝑚+1,𝑚+2,…. (3.5)
The result is sharp for every 𝑚, with the extremal function given by 𝑓(𝑧)𝛿=𝑧+𝑐_𝑚+1𝑧^𝑚+1. (3.6)

Next, if we put 𝜓(𝑧)=𝑧/(1−𝑧) in Theorem 2.1 for the function 𝑓 of the form (1.2) with 𝑐_𝑘=𝜌_𝑘(𝜆,𝛾,𝜂) and 𝛿=1−𝛾, then we obtain the following result of Murugusundaramoorthy et al. ([23], Theorem 2.1).

Corollary 3.4. If 𝑓 of the form (1.2) satisfies the condition (1.5) with 𝑐_𝑘=𝜌_𝑘(𝜆,𝛾,𝜂) and 𝛿=1−𝛾, then for 𝑧∈𝑈∶𝑓(𝑧)Re𝑓_𝑚(𝑧)≥𝜌_𝑚+1𝜆,𝛾,𝜂−1+𝛾𝜌_𝑚+1𝜆,𝛾,𝜂, (3.7) where 𝜌_𝑘≥𝜌𝜆,𝛾,𝜂1−𝛾if𝑘=2,3,…,𝑚,_𝑚+1𝜆,𝛾,𝜂if𝑘=𝑚+1,𝑚+2…. (3.8)
The result is sharp for every 𝑚, with the extremal function given by 𝑓(𝑧)=𝑧+1−𝛾𝜌_𝑚+1𝑧𝜆,𝛾,𝜂^𝑚+1. (3.9)

Again, if we set 𝜓(𝑧)=(𝑧/(1−𝑧))+((𝑧/(1−𝑧))−𝑧) in Theorem 2.1, then we obtain the following result of Porwal [28].

Corollary 3.5. If 𝑓 of the form (1.1) with 𝑏₁=0, satisfies the condition (1.4) with 𝑐_𝑘≥𝑐𝛿if𝑘=2,3,…,𝑚,_𝑚+1if𝑘=𝑚+1,𝑚+2…, (3.10) then 𝑓(𝑧)Re𝑓_𝑚(𝑧)≥𝑐_𝑚+1−𝛿𝑐_𝑚+1,∀𝑧∈𝑈. (3.11) The result (3.11) is sharp with the function given by (3.6).

Here we give some open problems for the readers.

In 2004, Owa et al. [24] studied the starlikeness and convexity properties on the partial sums 𝑓_𝑛(𝑧) and 𝑔_𝑛(𝑧) of the familiar Koebe function 𝑓(𝑧)=𝑧/(1−𝑧)² which is the extremal function for the class 𝑆^∗ of starlike functions in the open unit disk 𝑈 and the function 𝑔(𝑧)=𝑧/(1−𝑧) which is the extremal function for the class 𝐾 of convex functions in the open unit disk 𝑈, respectively. They also presented some illustrative examples by using Mathematica (Version 4.0). It is interesting to obtain analogues results on harmonic starlikeness and convexity properties of the partial sums of the harmonic Koebe function.

In 2003, Jahangiri et al. [30] studied the construction of sense-preserving, univalent, and close-to-convex harmonic functions by using of the Alexander integral transforms of certain analytic functions (which are starlike or convex of positive order). They construct a function 𝑔(𝑧)=𝑧+1−𝛼𝑧𝑘−𝛼^𝑘+𝛼2(𝑧)²+𝛼(1−𝛼)()(𝑘+1)(𝑘−𝛼𝑧)^𝑘+1, (3.12) which is sense-preserving, univalent, and close-to-convex harmonic in 𝑈, by using the result of Theorem 2 [30] and taking the following function: 𝑓(𝑧)=𝑧+1−𝛼𝑘()𝑧𝑘−𝛼^𝑘,(𝑘>1;0≤𝛼<1). (3.13) It is worthy to note that the function (3.13) is of the form (3.6) with 𝑐_𝑘=𝑘(𝑘−𝛼) and 𝛿=1−𝛼. Therefore, it is natural to ask that the results of [30] may be generalized for the function of the form (3.6).

Acknowledgment

The author is thankful to the referee for his valuable comments and observations which helped in improving the paper.