Coefficient Inequalities of Analytic Functions Related to Robertson Functions

Abstract

We introduce and study a subclass of analytic functions related to Robertson functions. Here we discuss the coefficient estimate for function in this class.

1. Introduction

Let 𝐴 be the class of functions 𝑓 of the form 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛 (1) which are analytic in the open unit disc 𝐸={𝑧∶|𝑧|<1}. Also let 𝑆^∗ and 𝐶 denote the well-known classes of starlike and convex functions, respectively.

For any two analytic functions 𝑓 given by (1) and 𝑔 with 𝑔(𝑧)=𝑧+^∞_𝑛=2𝑏_𝑛𝑧^𝑛,for𝑧∈𝐸, (2) the convolution (Hadamard product) is given by 𝑓⋆𝑔(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑏_𝑛𝑧^𝑛,for𝑧∈𝐸. (3) Using the concept of convolution, Ruscheweyh [1] introduced a differential operator 𝐷^𝛿 given by 𝐷^𝛿𝑓(𝑧)=𝑧(1−𝑧)^𝛿+1∗𝑓(𝑧)=𝑧+^∞_𝑛=2𝜑_𝑛(𝛿)𝑎_𝑛𝑧^𝑛,(𝛿>−1) (4) with 𝜑_𝑛(𝛿)=(𝛿+1)_𝑛−1()!,𝑛−1 (5) where (𝑥)_𝑛 is a Pochhammer symbol given as (𝑥)_𝑛=𝑥()⋯()1,𝑛=0,𝑥+1)(𝑥+2𝑥+𝑛−1,𝑛∈ℕ. (6) It is obvious that 𝐷⁰𝑓(𝑧)=𝑓(𝑧), 𝐷¹𝑓(𝑧)=𝑧𝑓^′(𝑧), and 𝐷^𝑛𝑓(𝑧)=𝑧𝑧^𝑛−1𝑓(𝑧)^(𝑛)𝑛!,∀𝛿=𝑛∈𝑁₀={0,1,2,…}. (7) The following identity can easily be established: (𝛿+1)𝐷^𝛿+1𝑓(𝑧)=𝛿𝐷^𝛿𝑓(𝑧)+𝑧𝐷^𝛿𝑓(𝑧)^′. (8) Now with the help of Ruscheweyh derivative, we define a class 𝑉𝐷_𝜆(𝛼,𝛽,𝑏,𝛿) of analytic functions as follows.

Definition 1. Let 𝑓(𝑧)∈𝐴. Then, 𝑓(𝑧)∈𝑉𝐷_𝜆(𝛼,𝛽,𝑏,𝛿), if and only if Re𝑒^𝑖𝜆21−𝑏+2𝑏𝐷^𝛿+1𝑓(𝑧)𝐷^𝛿𝑓(𝑧)2>𝛼𝑏𝐷^𝛿+1𝑓(𝑧)𝐷^𝛿𝑓(𝑧)−1+𝛽cos𝜆, (9) where 𝛼≥0, 0≤𝛽<1, 𝛿>−1, 𝜆 is real with |𝜆|<𝜋/2, and 𝑏∈ℂ⧵{0}.
By giving specific values to 𝛼, 𝛽, 𝜆, 𝑏, and 𝛿 in 𝑉𝐷_𝜆(𝛼,𝛽,𝑏,𝛿), we obtain many important subclasses studied by various authors in earlier papers, see for details [2–5], and list some of them as follows: (i)𝑉𝐷_𝜆(0,0,2,0)≡𝑆^∗_𝜆 and 𝑉𝐷_𝜆(0,0,1,1)≡𝐾_𝜆, studied by Spacek [6] and Robertson [7], respectively; for the advancement work, see [8, 9].(ii)𝑉𝐷₀(𝛼,𝛽,2,0)≡𝑆𝐷(𝛼,𝛽) and 𝑉𝐷₀(𝛼,𝛽,1,1)≡𝐾𝐷(𝛼,𝛽), studied by both Owa et al. and Shams et al. [10, 11].(iii)𝑉𝐷_𝜆(1,0,2,0)≡𝑈𝑆𝑃(𝜆), 𝑉𝐷_𝜆(1,0,1,1)≡UCSP(𝜆), introduced by Ravichandran et al. [12]. (iv)𝑉𝐷₀(𝛼,𝛽,𝑏,𝛿)≡𝑉𝐷(𝛼,𝛽,𝑏,𝛿), considered by Latha [13].(v)𝑉𝐷₀(0,𝛽,2,0)≡𝑆^∗(𝛽), 𝑉𝐷₀(0,𝛽,1,1)≡𝐶(𝛽), the well-known classes of starlike and convex functions of order 𝛽.
From the above special cases, we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.
We will assume throughout our discussion, unless otherwise stated, that 𝛼≥0, 0≤𝛽<1, 𝛿>−1, 𝜆 is real with |𝜆|<𝜋/2, and 𝑏∈ℂ⧵{0}.

2. Some Properties of the Class 𝑉𝐷_𝜆(𝛼,𝛽,𝑏,𝛿)

Theorem 2. If 𝑓(𝑧)∈𝑉𝐷_𝜆(𝛼,𝛽,𝑏,𝛿) with 0≤𝛼≤𝛽, then 𝑓(𝑧)∈𝑉𝐷_𝜆0,𝛽−𝛼.1−𝛼,𝑏,𝛿 (10)

Proof. Since Re𝑤≤|𝑤| for any complex number 𝑤, 𝑓(𝑧)∈𝑉𝐷_𝜆(𝛼,𝛽,𝑏,𝛿) implies that Re𝑒^𝑖𝜆21−𝑏+2𝑏𝐷^𝛿+1𝑓(𝑧)𝐷^𝛿𝑓(𝑧)2>𝛼𝑏𝐷^𝛿+1𝑓(𝑧)𝐷^𝛿𝑓(𝑧)−2𝑏𝑒+𝛽cos𝜆≥𝛼Re^𝑖𝜆21−𝑏+2𝑏𝐷^𝛿+1𝑓(𝑧)𝐷^𝛿𝑓(𝑧)+𝛽−𝛼cos𝜆 (11) which implies that Re𝑒^𝑖𝜆21−𝑏+2𝑏𝐷^𝛿+1𝑓(𝑧)𝐷^𝛿𝑓(𝑧)>𝛽−𝛼cos𝜆(),().1−𝛼𝑧∈𝐸 (12) And hence, we obtain the required result.
Put 𝜆=0, 𝑏=2, and 𝛿=0 in Theorem 2; we obtain the following result.

Corollary 3 (see [10]). If 𝑓(𝑧)∈𝑆𝐷(𝛼,𝛽) with 0≤𝛼≤𝛽, then 𝑓(𝑧)∈𝑆^∗𝛽−𝛼.1−𝛼 (13) Set 𝜆=0, 𝑏=1, and 𝛿=1 in Theorem 2; one has the following result.

Corollary 4 (see [10]). If 𝑓(𝑧)∈𝐾𝐷(𝛼,𝛽) with 0≤𝛼≤𝛽, then 𝑓(𝑧)∈𝐾𝛽−𝛼.1−𝛼 (14)

Theorem 5. If 𝑓(𝑧)∈𝑉𝐷_𝜆(𝛼,𝛽,𝑏,𝛿), then 𝑎₂≤|𝑏|𝜂||,𝑎1−𝛼_𝑛≤(𝛿+1)|𝑏|𝜂(𝑛−1)|1−𝛼|𝜑_𝑛(𝛿)^𝑛−2_𝑗=1(𝑏𝜂1+𝛿+1)||𝑗||,1−𝛼𝑛≥3, (15)(16) where 𝜂=1−𝛽²cos²𝜆+(1−𝛼)²sin²𝜆. (17)

Proof. We note that for 𝑓(𝑧)∈𝑉𝐷_𝜆(𝛼,𝛽,𝑏,𝛿), Re𝑒^𝑖𝜆21−𝑏+2𝑏𝐷^𝛿+1𝑓(𝑧)𝐷^𝛿𝑓(𝑧)>𝛽−𝛼(1−𝛼)cos𝜆,𝑧∈𝐸. (18) Let us define the function 𝑝(𝑧) by 𝑝(𝑧)=)×𝑒((1−𝛼_𝑖𝜆1−2/𝑏+(2/𝑏)𝐷^𝛿+1𝑓(𝑧)/𝐷^𝛿𝑓(𝑧)−𝛽−𝛼cos𝜆×1−𝛽cos𝜆+𝑖(1−𝛼)sin𝜆⁻¹. (19) Then, 𝑝(𝑧) is analytic in 𝐸 with 𝑝(0)=1 and Re𝑝(𝑧)>0. Let 𝑝(𝑧)=1+^∞_𝑛=1𝑝_𝑛𝑧^𝑛. (20) Then, (19) can be written as 21−𝑏+2𝑏𝐷^𝛿+1𝑓(𝑧)𝐷^𝛿𝑓(𝑧)()=1+1−𝛽cos𝜆+𝑖1−𝛼sin𝜆𝑒^𝑖𝜆(1−𝛼)^∞_𝑛=1𝑝_𝑛𝑧^𝑛, (21) and using (8), we have 2𝑒^𝑖𝜆(1−𝛼)()𝑧𝐷𝛿+1^𝛿𝑓(𝑧)^′−𝐷^𝛿𝑓(𝑧)=𝑏𝜂𝐷^𝛿𝑓(𝑧)^∞_𝑛=1𝑝_𝑛𝑧^𝑛 (22) which implies that 𝑒^𝑖𝜆𝑎_𝑛=𝑏𝜂(𝛿+1)2(1−𝛼)(𝑛−1)𝜑_𝑛(𝛿)×𝑝_𝑛−1+𝜑₂(𝛿)𝑎₂𝑝_𝑛−2+⋯+𝜑_𝑛−1(𝛿)𝑎_𝑛−1𝑝₁, (23) where we have used (4) and (5). Now applying the coefficient estimates |𝑝_𝑛|≤2 for Caratheodory function [14], we obtain 𝑎_𝑛≤(𝛿+1)|𝑏|𝜂(𝑛−1)|1−𝛼|𝜑_𝑛(𝛿)×1+𝜑₂(𝛿)𝑎₂+⋯+𝜑_𝑛−1(𝛿)𝑎_𝑛−1. (24)
For 𝑛=2, 𝑎₂≤|𝑏|𝜂|1−𝛼| (25) which proves (15).
For 𝑛=3, 𝑎₃≤(𝛿+1)|𝑏|𝜂2|1−𝛼|𝜑₃(𝛿)(𝑏𝜂1+𝛿+1)||||.1−𝛼 (26) Therefore, (16) holds for 𝑛=3. Assume that (16) is true for all 𝑛=3,4,…𝑘 and consider 𝑎_𝑘+1≤(𝛿+1)|𝑏|𝜂(𝑘+1−1)|1−𝛼|𝜑_𝑘+1(𝛿)×(𝑏𝜂1+𝛿+1)||||+(𝑏𝜂1−𝛼𝛿+1)||2||(𝑏𝜂1−𝛼1+𝛿+1)||||(𝑏𝜂1−𝛼+⋯+𝛿+1)||(𝑘−1)∣|1−𝛼|^𝑘−2_𝑗=1(𝑏𝜂1+𝛿+1)||𝑗||=(𝑏𝜂1−𝛼𝛿+1)||𝑘|1−𝛼|𝜑_𝑘+1(𝛿)^𝑘−1_𝑗=1(𝑏𝜂1+𝛿+1)||𝑗||.1−𝛼 (27) Thus, the result is true for 𝑛=𝑘+1, and hence by induction, (16) holds for all 𝑛≥3.
If we set 𝜆=0, 𝑏=2, and 𝛿=0 in Theorem 5, we get the result proved in [10].

Corollary 6. If 𝑓(𝑧)∈𝑆𝐷(𝛼,𝛽), then 𝑎₂≤21−𝛽||,𝑎1−𝛼_𝑛≤21−𝛽(𝑛−1)|1−𝛼|^𝑛−2_𝑗=121+1−𝛽𝑗||,().1−𝛼𝑛≥3 (28)

Remark 7. If we take 𝛼=0 in Corollary 6, we have 𝑎_𝑛≤1(𝑛−1)!^𝑛_𝑗=2𝑗−2𝛽,(𝑛≥2) (29) which was proved by Robertson [15].

By setting 𝜆=0, 𝑏=1, and 𝛿=1 in Theorem 5, we obtain the result in [10].

Corollary 8. If 𝑓(𝑧)∈𝐾𝐷(𝛼,𝛽), then 𝑎₂≤1−𝛽||,𝑎1−𝛼_𝑛≤21−𝛽𝑛(𝑛−1)|1−𝛼|^𝑛−2_𝑗=121+1−𝛽𝑗||,().1−𝛼𝑛≥3 (30)

Remark 9. Letting 𝛼=0 in Corollary 8, we have 𝑎_𝑛≤1𝑛!^𝑛_𝑗=2𝑗−2𝛽,(𝑛≥2) (31) given by Robertson [15].

Acknowledgment

The principle author would like to thank Prof. Dr. Ihsan Ali, Vice Chancellor Abdul Wali Khan University Mardan for providing excellent research facilities and financial support.