# Certain Subclasses of Multivalent Functions Defined by Higher-Order Derivative

Authors: Xiaofei Li; Deng Ding; Liping Xu; Chuan Qin; Songbo Hu
Journal: Journal of Function Spaces
 (2017)

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## Abstract

In this paper, we define and study some subclasses of multivalent analytic functions of higher order in the unit disc. These classes generalize some classes previously studied. We obtain coefficient inequalities, distortion theorems, extreme points, and integral mean inequalities. We derive some results as special cases.


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## Body

## 1. Introduction

Letp∈N=1,2,… and denote Ap as the class of multivalent functions of the form(1)fz=zp+∑n=p+1∞anzn,which are analytic in the open unit disk: (2)U=z∈C:z<1.For two parameters α∈-p,p and β≥0, function f(z)∈Ap is said to be in class UST(p,α,β) of p-valent β-uniformly star-like functions of order α in U, if and only if(3)Rzf′zfz-α≥βzf′zfz-p,where R(·) denotes taking the real part of argument. On the other hand, function f(z)∈Ap is said to be in class UCV(p,α,β) of p-valent β-uniformly convex functions of order α in U, if and only if(4)R1+zf′′zf′z-α≥β1+zf′′zf′z-p.We note from (3) and (4) that (5)fz∈UCVp,α,β⟺zf′zp∈USTp,α,β.The classes UST(p,α,β) and UCV(p,α,β) were introduced recently by Khairnar and More [1]. Various subclasses of analytic and univalent or multivalent functions were studied in many papers (see, e.g., [2–4]). Recently, Nishiwaki and Owa in [5] introduced two classes MD(α,β) consisting of all functions f(z)∈A1, which satisfy (6)Rzf′zfz-α<βzf′zfz-1,and ND(α,β) consisting of all functions f(z)∈A1, which satisfy (7)R1+zf′′zf′z-α<βzf′′zf′z,where α≥1 and β≤0. We notice from definitions of these classes that (8)fz∈NDα,β⟺zf′z∈MDα,β.For each f(z)∈Ap, it is easily seen upon differentiating both sides of (1) q times with respect to z that(9)fqz=δp,qzp-q+∑n=p+1∞δn,qanzn-q,where p>q∈N0=N∪0, and δ(p,q) denotes q-permutations of p objects; that is, (10)δp,q=p!p-q!.Letq,m∈N0 and p∈N such that p>q+m, and assume that (11)-δp-q,m≤α<δp-q,m,β≥0.Srivastava et al. in [6] introduced a subclass of the p-valent function class USm(p,q;α,β) consisting of functions f(z) of form (1), which satisfy the following analytic criterion: for all z∈U,(12)Rzmfq+mzfqz-α≥βzmfq+mzfqz-δp-q,m.Recently, many papers have discussed such aspects of analytic univalent or multivalent functions. For example, Aouf et al. in [7, 8] discussed a subclass of p-valent analytic function with negative coefficient by using higher-order derivatives and investigated many properties of distortion theorems, closure theorems, modified Hadamard products, and radii of close-to-convexity, starlikeness, and convexity. K. I. Noor and M. A. Noor in [9] defined some subclasses of analytic functions related to k-uniformly close-to-convex functions of higher order and studied the following: rate of growth of coefficients, inclusion relations, radius problems, and necessary conditions for univalency. Seoudy in [10] defined a class of p-valent functions defined by certain linear operator and obtained subordination and inclusion properties.Following Srivastava et al. [6] and Nishiwaki and Owa [5], we define a new subclass MDm(p,q;α,β) of multivalent functions involving higher-order derivative.Definition 1.
Letq,m∈N0 and p∈N such that p>q+m, and α and β, two real numbers, satisfy that α≥δ(p-q,m) and β≤0. Function f(z)∈MDm(p,q;α,β), if and only if f(z)∈Ap and satisfies the following inequality:(13)Rzmfq+mzfqz-α<βzmfq+mzfqz-δp-q,m.From the above definition, it is clear thatMD1(1,0;α,β)=MD(α,β). In this paper, we obtain several properties including the coefficient inequalities, distortion theorems, extreme points, and integral means inequalities for this subclass MDm(p,q;α,β) of multivalent functions involving higher-order derivative.


## 2. Coefficient Inequalities

We derive sufficient conditions forf(z) which are given by using coefficient inequalities.Theorem 2.
Letf(z) be a function of form (1). If the coefficients of f(z) satisfy(14)∑n=p+1∞Φn,p,q,m,α,βan≤δp,qα-2δp-q,m-α,where(15)Φn,p,q,m,α,β=δn,qδn-q,m+δp-q,m-α+δn-q,m-δp-q,m-α-2βδn-q,m-δp-q,m,then, f(z) is in class MDm(p,q;α,β).Proof.
Suppose that inequality (14) holds, and denote (16)Fz=zmfq+mzfqz-α-βzmfq+mzfqz-δp-q,m.From the definition, we can verify that if (17)Fz+δp-q,mFz-δp-q,m<1,then f(z)∈MDm(p,q;α,β). In fact, denoting(18)Ψ±z;f,m,p,q,α,β=zmfq+mz-αfqz±δp-q,mfqz-βeiθzmfq+mz-δp-q,mfqz,Σ±z;m,p,q,α=∑n=p+1∞δn,qδn-q,m±δp-q,m-αanzn-p,Ξ±m,p,q,α=∑n=p+1∞δn,qδn-q,m±δp-q,m-α·an,we have(19)Fz+δp-q,mFz-δp-q,m=Ψ+z;f,m,p,q,α,βΨ-z;f,m,p,q,α,β=δp,q2δp-q,m-α+Σ+z;m,p,q,α-βeiθΣ-z;m,p,q,0-αδp,q-Σ-z;m,p,q,α+βeiθΣ-z;m,p,q,0≤δp,q2δp-q,m-α+Ξ+m,p,q,α-βΞ-m,p,q,0αδp,q-Ξ-m,p,q,α+βΞ-m,p,q,0.Here, we use technology f(z)=eiθfz. If (14) satisfies, we drive that the last expression above is bounded by 1 which implies f(z)∈MDm(p,q;α,β). Thus, the proof of Theorem 2 is completed.Example 3.
Functionf(z) given by (20)fz=zp+∑n=p+1∞δp,qα-2δp-q,m-αγ+p+1ξnn+γn+γ+1Φn,p,q,m,α,βznbelongs to class MDm(p,q;α,β) for γ>-(p+1) and ξn∈C with ξn=1.As a special case of Theorem2, as in [2], we can obtain the following corollary.Corollary 4.
Functionf(z)∈A1 is in class MD(α,β), if (21)∑n=2∞n+1-α+n-1-α-2βn-1·an≤α-2-α.In view of Theorem2, we introduce subclass MDm∗(p,q;α,β) which consists of functions of the form(22)fz=zp+∑n=p+1∞anzn,whose Taylor-Maclaurin coefficients an are nonnegative and satisfy inequality (14). By the coefficient inequalities for classes MDm∗(p,q;α,β), we have the following theorem.Theorem 5.
Ifβ1≤β2≤0, then (23)MDm∗p,q;α,β1⊂MDm∗p,q;α,β2.SinceMD1(1,0;α,β)=MD(α,β), we get the following corollary, which is a theorem in [2].Corollary 6.
Ifβ1≤β2≤0, then (24)MDα,β1⊂MDα,β2.


## 3. Distortion Theorems

Lemma 7.
Iff(z)∈MDm∗(p,q;α,β), then there exists p0∈N such that(25)∑n=p0+1∞an≤Ap0,where(26)Ap0=δp,qα-2δp-q,m-α-∑n=p+1p0Φn,p,q,m,α,βanΦp0+1,p,q,m,α,β,and Φ(n,p,q,m,α,β) is given in (15).Proof.
From the definition ofΦ(n,p,q,m,α,β), there exists p0∈N such that function Φ(n,p,q,m,α,β) is increasing with respect to n when n>p0. According to Theorem 2, we have(27)∑n=p0+1∞Φn,p,q,m,α,βan≤δp,qα-2δp-q,m-α-∑n=p+1p0Φn,p,q,m,α,βan.From(28)Φp0+1,p,q,m,α,β∑n=p0+1∞an≤∑n=p0+1∞Φn,p,q,m,α,βan,we have(29)Φp0+1,p,q,m,α,β∑n=p0+1∞an≤δp,qα-2δp-q,m-α-∑n=p+1p0Φn,p,q,m,α,βan.This implies that inequality (25) holds.Using the same argument, we obtain the following inequality.Lemma 8.
Iff(z)∈MDm∗(p,q;α,β), then there exists p0∈N such that(30)∑n=p0+1∞nan≤Bp0,where(31)Bp0=p0+1δp,qα-2δp-q,m-α-∑n=p+1p0Φn,p,q,m,α,βanΦp0+1,p,q,m,α,β,and Φ(n,p,q,m,α,β) is defined by (15).Theorem 9.
Letf(z) be a function in class MDm∗(p,q;α,β). Then, for z=r<1, (32)fz≤rp+∑n=p+1p0anrn+Ap0rp0+1,fz≥rp-∑n=p+1p0anrn-Ap0rp0+1,where Ap0 and Bp0 are given in Lemmas 7 and 8, respectively.Proof.
Letf(z) be a function of form (22). For z=r<1, using Lemma 7, we have (33)fz≤zp+∑n=p+1p0an·zn+∑n=p0+1∞an·zn≤zp+∑n=p+1p0an·zn+zp0+1∑n=p0+1∞an≤rp+∑n=p+1p0anrn+Ap0rp0+1,fz≥zp-∑n=p+1p0an·zn-∑n=p0+1∞an·zn≥zp-∑n=p+1p0an·zn-zp0+1∑n=p0+1∞an≥rp-∑n=p+1p0anrn-Ap0rp0+1.Using the same argument, we can prove the following result.Theorem 10.
Letf(z) be a function in class MDm∗(p,q;α,β). Then, for z=r<1, (34)f′z≤prp-1+∑n=p+1p0nanrn-1+Bp0rp0,f′z≥prp-1-∑n=p+1p0nanrn-1-Bp0rp0,where An and Bn are given in Lemmas 7 and 8, respectively.


## 4. Extreme Points

Theorem 11.
Letfp(z)=zp and, for each n=p+1,p+2,…, define(35)fnz=zp+δp,qα-2δp-q,m-αΦn,p,q,m,α,βzn,where Φ(n,p,q,m,α,β) is defined by (15). Then f(z)∈MDm∗(p,q;α,β) if and only if it can be expressed in the form (36)fz=∑n=p∞λnfnz,where λn≥0 for all n=p,p+1,…, and ∑n=p∞λn=1.Proof.
Suppose that(37)fz=∑n=p∞λnfnz=λpfpz+∑n=p+1∞λnfnz=zp+∑n=p+1∞λnδp,qα-2δp-q,m-αΦn,p,q,m,α,βzn.Then,(38)∑n=p+1∞Φn,p,q,m,α,βλnδp,qα-2δp-q,m-αΦn,p,q,m,α,β=∑n=p+1∞λnδp,qα-2δp-q,m-α=δp,qα-2δp-q,m-α1-λp≤δp,qα-2δp-q,m-α.Thus, it follows from Theorem 2 that f(z)∈MDm∗(p,q;α,β).
Conversely, suppose thatf(z)∈MDm∗(p,q;α,β). Since (39)an≤δp,qα-2δp-q,m-αΦn,p,q,m,α,β,n=p+1,p+2,…,we denote(40)λn=Φn,p,q,m,α,βδp,qα-2δp-q,m-αan,n=p+1,p+2,….And λp=1-∑n=p+1∞λn. By a simple calculation, we get f(z)∈MDm∗(p,q;α,β).Corollary 12.
The extreme points ofMDm∗(p,q;α,β) are functions fp(z)=zp and (41)fnz=zp+δp,qα-2δp-q,m-αΦn,p,q,m,α,βzn,for each n=p+1,p+2,….


## 5. Integral Means Inequalities

Assume that two functionsf(z) and g(z) are analytic in U. We say that f(z) is subordinate to g(z), written as f(z)≺g(z), if there exists an analytic function w(z) in U with w(0)=1 and wz<1 such that f(z)=g(w(z)).Lemma 13 (see [11]).
Iff(z) and g(z) are analytic in U with f(z)≺g(z), then, for μ>0 and z=reiθ(0<r<1),(42)∫02πfzμdθ≤∫02πgzμdθ.Theorem 14.
Letf(z)∈MDm∗(p,q;α,β) and fn(z) be given by (35). Suppose that(43)∑n=p+1∞an≤Φn,p,q,m,α,βδp,qα-2δp-q,m-α.If there exists function w(z), z∈U, that satisfied the condition (44)wz=Φn,p,q,m,α,βδp,qα-2δp-q,m-α∑n=p+1∞anzn-p1/n-p,then, for z=reiθ,0<r<1, we have (45)∫02πfzμdθ≤∫02πfnzμdθ.Proof.
In order to obtain the result, it is necessary to prove the following inequality:(46)∫02π1+∑n=p+1∞anzn-pμdθ≤∫02π1+δp,qα-2δp-q,m-αΦn,p,q,m,α,βzn-pμdθ.From Lemma 13, it is sufficient to verify the subordination: (47)1+∑n=p+1∞anzn-p≺1+δp,qα-2δp-q,m-αΦn,p,q,m,α,βzn-p.Thus, there exists an analytic function w(z) in C such that (48)1+∑n=p+1∞anzn-p=1+δp,qα-2δp-q,m-αΦn,p,q,m,α,βwzn-p.We find that(49)wz=Φn,p,q,m,α,βδp,qα-2δp-q,m-α∑n=p+1∞anzn-p1/n-p,which readily yields w(0)=0 and (50)wz1/n-p=Φn,p,q,m,α,βδp,qα-2δp-q,m-α∑n=p+1∞anzn-p≤Φn,p,q,m,α,βδp,qα-2δp-q,m-α∑n=p+1∞anzn-p≤Φn,p,q,m,α,βδp,qα-2δp-q,m-αz∑n=p+1∞an≤z<1.This means that the hypotheses of w(z) are satisfied and the theorem is proved.



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