Certain Inequalities of Multivalent Analytic Functions with Missing Coefficients

Abstract

The purpose of the present paper is to derive the radius of starlikeness for certain p-valent functions with missing coefficients. The results obtained here are sharp.

1. Introduction

Let 𝐴_𝑝(𝑛) denote the class of functions of the form 𝑓(𝑧)=𝑧^𝑝+^∞_𝑘=𝑛𝑎_𝑝+𝑘𝑧^𝑝+𝑘𝑝,𝑛∈𝑁={1,2,3,…}, (1) which are 𝑝-valent analytic in the open unit disk 𝑈={𝑧∶𝑧∈𝐶and|𝑧|<1}. A function 𝑓(𝑧)∈𝐴_𝑝(𝑛) is said to be 𝑝-valently starlike of order 𝜌 in 𝑈 if it satisfies Re𝑧𝑓^′(𝑧)𝑓(𝑧)>𝜌0≤𝜌<𝑝;𝑧∈𝑈. (2)

For functions 𝑓(𝑧) and 𝑔(𝑧) analytic in 𝑈, we say that 𝑓(𝑧) is subordinate to 𝑔(𝑧) in 𝑈, and we write 𝑓(𝑧)≺𝑔(𝑧)(𝑧∈𝑈), if there exists an analytic function 𝑤(𝑧) in 𝑈 such that |𝑤(𝑧)|≤|𝑧|,𝑓(𝑧)=𝑔(𝑤(𝑧))(𝑧∈𝑈). (3) Furthermore, if the function 𝑔(𝑧) is univalent in 𝑈, then 𝑓(𝑧)(𝑧)(0)(0),𝑓(𝑈)(𝑈).≺𝑔)(𝑧∈𝑈⟺𝑓=𝑔⊂𝑔 (4)

A number of results for 𝑝-valently starlike functions have been obtained by several authors (see, e.g., [1–7]). In this note, we shall derive the radius of starlikeness for certain 𝑝-valent functions with missing coefficients.

2. Main Results

Our main result is the following.

Theorem 1. Let 𝑓(𝑧) belong to the class 𝐴_𝑝(𝑛) and satisfy 𝑓(𝑧)𝑧^𝑝≺1+𝑧1−𝑧^𝛾0<𝛾≤1;𝑧∈𝑈. (5) Then, )𝑓(𝑧)Re(1−𝛿𝑧^𝑝1/𝛾+𝛿𝑧𝑓^′(𝑧)𝑓(𝑧)|𝑧|>𝜌<𝑟_𝑛𝑝,𝛾,𝛿,𝜌, (6) where 0<𝛿≤1, 0≤𝜌<1 and 𝑟_𝑛(𝑝,𝛾,𝛿,𝜌) is the smallest root in (0,1) of the equation 1+𝜌−𝑝+1𝛿𝑟^2𝑛−21−𝛿+𝑛𝛿𝛾𝑟^𝑛+1−𝜌+𝑝−1𝛿=0. (7) The result is sharp.

Proof. From (5), we can write that 𝑓(𝑧)𝑧^𝑝1/𝛾=1+𝑧^𝑛𝜑(𝑧)1−𝑧^𝑛𝜑(𝑧), (8) where 𝜑(𝑧) is analytic and |𝜑(𝑧)|≤1 in 𝑈. Differentiating both sides of (8) logarithmically, we arrive at 𝑧𝑓^′(𝑧)𝑓(𝑧)=𝑝+2𝑛𝛾𝑧^𝑛𝜑(𝑧)1−𝑧^𝑛𝜑(𝑧)²+2𝛾𝑧^𝑛+1𝜑^′(𝑧)1−𝑧^𝑛𝜑(𝑧)²(𝑧∈𝑈). (9) Put |𝑧|=𝑟<1 and (𝑓(𝑧)/𝑧^𝑝)^1/𝛾=𝑢+𝑖𝑣  (𝑢,𝑣∈𝑅). Then, (8) implies that 𝑧^𝑛𝜑(𝑧)=𝑢−1+𝑖𝑣,𝑢+1+𝑖𝑣1−𝑟^𝑛1+𝑟^𝑛≤𝑢≤1+𝑟^𝑛1−𝑟^𝑛. (10)(11) With the help of the Carathéodory inequality: 𝜑^′(𝑧≤𝜑(𝑧)1−)²1−𝑟², (12) it follows from (10) that )𝑓(𝑧)Re(1−𝛿𝑧^𝑝1/𝛾+𝛿𝑧𝑓^′(𝑧)𝑓(𝑧)≥()𝑧1−𝛿𝑢+𝑝𝛿+2𝑛𝛿𝛾Re^𝑛𝜑(𝑧)1−𝑧^𝑛𝜑(𝑧)²𝑧−2𝛿𝛾^𝑛+1𝜑^′(𝑧)1−𝑧^𝑛𝜑(𝑧)²≥(1−𝛿)𝑢+𝑝𝛿+𝑛𝛿𝛾2𝑢𝑢−𝑢²+𝑣²+𝛿𝛾2(𝑢−1)²+𝑣²−𝑟^2𝑛(𝑢+1)²+𝑣²𝑟^𝑛−11−𝑟²𝑢²+𝑣^21/2=𝐹_𝑛(),𝜕𝑢,𝑣say𝐹𝜕𝑣^𝑛(𝑢,𝑣)=𝛿𝛾𝑣𝐺_𝑛(𝑢,𝑣), (13)(14) where 0<𝑟<1, 0<𝛿≤1 and 𝐺_𝑛(𝑢,𝑣)=𝑛𝑢𝑢²+𝑣²²+1−𝑟^2𝑛𝑟^𝑛−11−𝑟²𝑢²+𝑣^21/2+𝑟^2𝑛(𝑢+1)²+𝑣²−(𝑢−1)²+𝑣²2𝑟^𝑛−11−𝑟²𝑢²+𝑣^23/2. (15) From (10), we can see that 𝑟^2𝑛(𝑢+1)²+𝑣²−(𝑢−1)²+𝑣²≥0. (16) Thus, we have from (10), (11), and (15) that 𝐺_𝑛(𝑢,𝑣)>0. (17) Since 𝐹_𝑛(𝑢,𝑣) is a even function of 𝑣, from (13), (14), and (15), we see that 𝐹_𝑛(𝑢,𝑣)≥𝐹_𝑛()=()𝑢,01−𝛿𝑢+𝑝𝛿+𝑛𝛿𝛾21𝑢−𝑢+𝛿𝛾2𝑟^𝑛−11−𝑟²×1−𝑟^2𝑛1𝑢+𝑢−21+𝑟^2𝑛. (18)
Let us now calculate the minimum value of 𝐹_𝑛(𝑢,0) on the closed interval [(1−𝑟^𝑛)/(1+𝑟^𝑛),(1+𝑟^𝑛)/(1−𝑟^𝑛)]. Noting that 1−𝑟^2𝑛𝑟^𝑛−11−𝑟²≥𝑛 (19)(see [8]) and lower bound in (11), we deduce from (18) that 𝑑𝐹𝑑𝑢^𝑛()+𝑢,0=1−𝛿𝛿𝛾21−𝑟^2𝑛𝑟^𝑛−11−𝑟²−1+𝑛𝑢²1−𝑟^2𝑛𝑟^𝑛−11−𝑟²+−𝑛≥1−𝛿𝛿𝛾21−𝑟^2𝑛𝑟^𝑛−11−𝑟²−+𝑛1+𝑟^𝑛1−𝑟^𝑛21−𝑟^2𝑛𝑟^𝑛−11−𝑟²−𝑛=1−𝛿+2𝛿𝛾𝐼_𝑛(𝑟)(1−𝑟^𝑛)², (20) where 𝐼_𝑛(𝑟)=𝑛21+𝑟^2𝑛−𝑟1+𝑟²+⋯+𝑟^2𝑛−2. (21) Also, 𝐼^′_𝑛(𝑟)=𝑛²𝑟^2𝑛−1−1+3𝑟²+⋯+(2𝑛−1)𝑟^2𝑛−2. (22)𝐼^′₁(𝑟)=𝑟−1<0. Suppose that 𝐼^′_𝑛(𝑟)<0. Then, 𝐼^′_𝑛+1(𝑟)=(𝑛+1)²𝑟^2𝑛+1−(2𝑛+1)𝑟^2𝑛−1+3𝑟²+⋯+(2𝑛−1)𝑟^2𝑛−2<𝑛²𝑟^2𝑛−1+3𝑟²+⋯+(2𝑛−1)𝑟^2𝑛−2<𝐼^′_𝑛(𝑟)<0. (23) Hence, by virtue of the mathematical induction, we have 𝐼^′_𝑛(𝑟)<0 for all 𝑛∈𝑁 and 0≤𝑟<1. This implies that 𝐼_𝑛(𝑟)>𝐼_𝑛(1)=0(𝑛∈𝑁;0≤𝑟<1). (24) In view of (20) and (24), we see that when 𝑢∈[(1−𝑟^𝑛)/(1+𝑟^𝑛),(1+𝑟^𝑛)/(1−𝑟^𝑛)], then 𝑑𝐹𝑑𝑢^𝑛(𝑢,0)>0. (25) Further, it follows from (13), (18), and (25) that )𝑓(𝑧)Re(1−𝛿𝑧^𝑝1/𝛾+𝛿𝑧𝑓^′(𝑧)𝑓(𝑧)−𝜌≥𝐹_𝑛1−𝑟^𝑛1+𝑟^𝑛=(),0−𝜌1−𝛿1−𝑟^𝑛1+𝑟^𝑛+𝛿𝑝−2𝑛𝛾𝑟^𝑛−𝑝𝑟^2𝑛1−𝑟^2𝑛=𝐽−𝜌_𝑛(𝑟)1−𝑟^2𝑛, (26) where 0≤𝜌<1 and 𝐽_𝑛(𝑟)=1+𝜌−𝑝+1𝛿𝑟^2𝑛−21−𝛿+𝑛𝛿𝛾𝑟^𝑛+1−𝜌+𝑝−1𝛿. (27) Note that 𝐽_𝑛(0)=1−𝜌+(𝑝−1)𝛿>0 and 𝐽_𝑛(1)=−2𝑛𝛿𝛾<0. If we let 𝑟_𝑛(𝑝,𝛾,𝛿,𝜌) denote the smallest root in (0,1) of the equation 𝐽_𝑛(𝑟)=0, then (26) yields the desired result (6).
To see that the bound 𝑟_𝑛(𝑝,𝛾,𝛿,𝜌) is the best possible one, we consider the function 𝑓(𝑧)=𝑧^𝑝1+𝑧^𝑛1−𝑧^𝑛𝛾. (28) It is clear that for 𝑧=𝑟∈(𝑟_𝑛(𝑝,𝛾,𝛿,𝜌),1), (1−𝛿)𝑓(𝑟)𝑟^𝑝1/𝛾+𝛿𝑟𝑓^′(𝑟)𝑓(𝑟)𝐽−𝜌=^𝑛(𝑟)1−𝑟^2𝑛<0, (29) which shows that the bound 𝑟_𝑛(𝑝,𝛾,𝛿,𝜌) cannot be increased.
Setting 𝛿=1, Theorem 1 reduces to the following result which gives the radius of starlikeness for certain 𝑝-valent analytic functions with missing coefficients.

Corollary 2. Let 𝑓(𝑧) satisfy the condition (5) and 0≤𝜌<1. Then, 𝑓(𝑧) is starlike of order 𝜌 in |𝑧|<𝑛𝛾²+𝑝−𝜌^21/2−𝑛𝛾𝑝−𝜌^1/𝑛. (30) The result is sharp.

Acknowledgment

The authors would like to express sincere thanks to the referees for careful reading and suggestions which helped them to improve the paper.