Properties of Certain Subclass of Multivalent Functions with Negative Coefficients

Abstract

Making use of a linear operator, which is defined by the Hadamard product, we introduce and study a subclass 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼) of the class 𝐴⁺(𝑝). In this paper, we obtain the coefficient inequality, distortion theorem, radius of convexity and starlikeness, neighborhood property, modified convolution properties of this class. Furthermore, an application of fractional calculus operator is given. The results are presented here would provide extensions of some earlier works. Several new results are also obtained.

1. Introduction

Let 𝐴⁺(𝑝) denote the class of functions 𝑓(𝑧)=𝑧^𝑝−^∞_𝑘=1𝑎_𝑘+𝑝𝑧^𝑘+𝑝𝑝∈𝑁={1,2,…} (1.1) which are analytic and 𝑝-valent in the open unit disc 𝑈={𝑧∶|𝑧|<1} on the complex plane 𝐶.

We denote by 𝑆^∗_𝑝(𝐴,𝐵) and 𝐾_𝑝(𝐴,𝐵)(−1≤𝐵<𝐴≤1) the subclass of 𝑝-valent starlike functions and the subclass of 𝑝-valent convex functions, respectively, that is, (see for details [1, 2]) 𝑆^∗_𝑝(𝐴,𝐵)=𝑓(𝑧)∈𝐴𝑝∶𝑧𝑓^′(𝑧)(𝑧)≺𝑝𝑓1+𝐴𝑧(,𝐾1+𝐵𝑧𝑧∈𝑈;−1≤𝐵<𝐴≤1)_𝑝()=𝑓(𝑧)𝑝∶1𝐴,𝐵∈𝐴𝑝+𝑧𝑓^′′(𝑧)𝑝𝑓^′(𝑧)≺1+𝐴𝑧(.1+𝐵𝑧𝑧∈𝑈;−1≤𝐵<𝐴≤1) (1.2)

A function 𝑓∈𝐴⁺(𝑝) is said to be analytic starlike of order 𝜉 if it satisfies Re𝑧𝑓^′(𝑧)𝑝𝑓(𝑧)>𝜉 (1.3) for some 𝜉(0≤𝜉<1) and for all 𝑧∈𝑈.

Further, a function 𝑓∈𝐴⁺(𝑝) is said to be analytic convex of order 𝜉 if it satisfies 1Re𝑝+𝑧𝑓^′′(𝑧)𝑝𝑓^′(𝑧)>𝜉 (1.4) for some 𝜉(0≤𝜉<1) and for all 𝑧∈𝑈.

For 𝑓(𝑧)=𝑧^𝑝+∑^∞_𝑘=1𝑎_𝑘+𝑝𝑧^𝑘+𝑝 and 𝑔(𝑧)=𝑧^𝑝+∑^∞_𝑘=1𝑏_𝑘+𝑝𝑧^𝑘+𝑝, the Hadamard product (or convolution) is defined by 𝑓∗𝑔(𝑧)=𝑧^𝑝+^∞_𝑘=1𝑎_𝑘+𝑝𝑏_𝑘+𝑝𝑧^𝑘+𝑝=𝑔∗𝑓(𝑧). (1.5)

The linear operator 𝐿_𝑝(𝑎,𝑐) is defined as follows (see Saitoh [3]): 𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)=𝜑_𝑝(𝑎,𝑐;𝑧)∗𝑓(𝑧)𝑓(𝑧)∈𝐴𝑝, (1.6) and 𝜑_𝑝(𝑎,𝑐;𝑧) is defined by 𝜑_𝑝(𝑎,𝑐;𝑧)=𝑧^𝑝+^∞_𝑘=1(𝑎)_𝑘(𝑐)_𝑘𝑧^𝑘+𝑝𝑎∈𝑅;𝑐⧵𝑧⁻₀;𝑧⁻₀∶={0,−1,−2,…}, (1.7) where (𝑣)_𝑘 is the pochharmmer symbol defined (in terms of the Gamma function) by (𝜈)_𝑛=Γ(𝜈+𝑛)Γ(𝜈)=𝜈()⋯()1,𝑛=0,𝜈+1𝜈+𝑛−1,𝑛∈𝑁. (1.8)

The operator 𝐿_𝑝(𝑎,𝑐) was studied recently by Srivastava and Patel [4]. It is easily verified from (1.6) and (1.7) that 𝑧𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′=𝑎𝐿_𝑝(𝑎+1,𝑐)𝑓(𝑧)−𝑎−𝑝𝐿_𝑝(𝑎,𝑐)𝑓(𝑧). (1.9)

Moreover, for 𝑓(𝑧)∈𝐴(𝑝),𝐿_𝑝(𝑎,𝑎)𝑓(𝑧)=𝑓(𝑧),𝐿_𝑝𝑝+1,𝑝𝑓(𝑧)=𝑧𝑓^′(𝑧)𝑝,𝐿_𝑝𝑛+𝑝,1𝑓(𝑧)=𝐷^{𝑛+𝑝−1}𝑓(𝑧)𝑛>−𝑝, (1.10) where 𝐷^{𝑛+𝑝−1}𝑓(𝑧) denotes the Ruscheweyh derivative of a function 𝑓(𝑧)∈𝐴(𝑧) of order 𝑛+𝑝−1 (see [5]).

Aouf, Silverman and Srivastava [6] introduced the class of 𝑃⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆) by use of linear operator 𝐿_𝑝(𝑎,𝑐), further investigated the properties of this class. In [7], SokóŁ investigated several properties of the linear Aouf-Silverman-Srivastava operator and furthermore obtained the corresponding characterizations of multivalent analytic functions which were studied by Aouf et al. in paper [6]. The properties of multivalent functions with negative coefficients were studied in [8–10]. References [11, 12] gave the results of the univalent function with negative coefficients.

In this paper, we will use operator 𝐿_𝑝(𝑎,𝑐) to define a new subclass of 𝐴⁺(𝑝) as follows.

For 𝑝∈𝑁,𝑎>0,𝑐>0,𝛼≥0 and for the parameters 𝜆,𝐴,and𝐵 such that −1≤𝐵<𝐴≤1,−1≤𝐵<0,0≤𝜆<𝑝, (1.11) we say that a function 𝑓(𝑧)∈𝐴⁺(𝑝) is in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼) if it satisfies the following subordination: 1𝐿𝑝−𝜆^𝑝(𝑎,𝑐)𝑓(𝑧)^′𝑧^𝑝−1𝐿−𝛼^𝑝(𝑎,𝑐)𝑓(𝑧)^′𝑝𝑧^𝑝−1−1−𝜆≺1+𝐴𝑧()1+𝐵𝑧𝑧∈𝑈, (1.12) or equivalently, if the following inequality holds true: 𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑧^𝑝−1−𝛼𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑝𝑧^𝑝−1−1−𝑝𝐵𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑧^𝑝−1−𝛼𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑝𝑧^𝑝−1−1−𝑝𝐵+(𝐴−𝐵)𝑝−𝜆<1. (1.13)

From the above definition, we can imply that the function class 𝑃⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆) in [6] is the special case of 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼) in our present paper because 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,0)=𝑃⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆).

Since 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,0)=𝑃⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆), then, like [6], we have the following subclasses which were studied in many earlier works:(1)𝑌⁺_𝑛+𝑝,1(−1,1;𝑝,𝜆,0)=𝑄_{𝑛+𝑝−1}(𝜆)(0≤𝜆<1;𝑛>−𝑝;𝑝∈𝑁)(Aouf and Darwish [8]),(2)𝑌⁺_𝑎,𝑎(𝐴,𝐵;𝑝,0,0)=𝑃^∗(𝑝,𝐴,𝐵)(Shukla and Dashrath [9]),(3)𝑌⁺_𝑎,𝑎(1,−1;𝑝,𝜆,0)=𝐹_𝑝(1,𝜆)(0≤𝜆<𝑝;𝑝∈𝑁)(Lee et al. [10]),(4)𝑌⁺_𝑎,𝑎(−𝛽,𝛽;1,𝜆,0)=𝑃^∗(𝜆,𝛽)(0≤𝜆<1;0<𝛽≤1)(Gupta and Jain [11]),(5)𝑌⁺_𝑛+𝑝,1(−1,1;1,𝜆,0)=𝑄_𝑛(𝜆)(𝑛∈𝑁₀∶=𝑁∪{0};0≤𝜆<1) (Uralegaddi and Sarangi [12]).

The purpose of this paper is to give various properties of class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼). We extend the results of basic paper [6].

2. Necessary and Sufficient Condition for 𝑓(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼)

Theorem 2.1. Let the function 𝑓(𝑧)∈𝐴⁺(𝑝) be given by (1.1), then 𝑓(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼) if and only if ^∞_𝑘=1𝜙_𝑘𝑝,𝛼;𝐵,𝑎,𝑐𝑎_𝑘+𝑝≤(𝐴−𝐵), (2.1) where 𝜙_𝑘=()(𝑎)𝑝,𝛼;𝐵,𝑎,𝑐𝑝+𝛼1−𝐵𝑘+𝑝_𝑘𝑝𝑝−𝜆(𝑐)_𝑘. (2.2)

Proof. For the sufficient condition, let 𝑓(𝑧)=𝑧^𝑝−∑^∞_𝑘=1|𝑎_𝑘+𝑝|𝑧^𝑘+𝑝, we have 𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑧^𝑝−1−𝛼𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑝𝑧^𝑝−1−1−𝑝𝐵𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑧^𝑝−1−𝛼𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑝𝑧^𝑝−1−(=𝑝𝐿𝑝𝐵+𝐴−𝐵)𝑝−𝜆_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝛼𝑒^𝑖𝜃𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝑝𝑧^𝑝−1−𝑝²𝑧^𝑝−1𝐵𝑝𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝛼𝑒^𝑖𝜃𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝑝𝑧^𝑝−1−𝑝𝑝𝐵+(𝐴−𝐵)𝑝−𝜆𝑧^𝑝−1, (2.3) then 𝑝𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝛼𝑒^𝑖𝜃𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝑝𝑧^𝑝−1−𝑝²𝑧^𝑝−1−𝐵𝑝𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝛼𝑒^𝑖𝜃𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝑝𝑧^𝑝−1−𝑝𝑝𝐵+(𝐴−𝐵)𝑝−𝜆𝑧^𝑝−1=−^∞_𝑘=1𝑝𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘𝑧^{𝑘+𝑝−1}−𝛼𝑒^𝑖𝜃−^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘𝑧^{𝑘+𝑝−1}−−^∞_𝑘=1𝐵𝑝𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘𝑧^{𝑘+𝑝−1}−𝛼𝐵𝑒^𝑖𝜃−^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘𝑧^{𝑘+𝑝−1}−𝑝(𝐴−𝐵)𝑝−𝜆𝑧^𝑝−1≤^∞_𝑘=1𝑝𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘|𝑧|^{𝑘+𝑝−1}+𝛼𝑒^𝑖𝜃∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘|𝑧|^{𝑘+𝑝−1}+|𝐵|^∞_𝑘=1𝑝𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘|𝑧|^{𝑘+𝑝−1}+𝛼|𝐵|^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘|𝑧|^{𝑘+𝑝−1}−𝑝(𝐴−𝐵)𝑝−𝜆|𝑧|^𝑝−1=𝑝+𝛼^∞_𝑘=1𝑘+𝑝(1−𝐵)𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘−𝑝(𝐴−𝐵)𝑝−𝜆(𝑧∈𝜕𝑈). (2.4) By the maximum modulus theorem, for any 𝑧∈𝑈, we have 𝑝𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝛼𝑒^𝑖𝜃𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝑝𝑧^𝑝−1−𝑝²𝑧^𝑝−1−𝐵𝑝𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝛼𝑒^𝑖𝜃𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′−𝑝𝑧^𝑝−1−𝑝𝑝𝐵+(𝐴−𝐵)𝑝−𝜆𝑧^𝑝−1≤𝑝+𝛼^∞_𝑘=1𝑘+𝑝(1−𝐵)𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘()=−𝑝𝐴−𝐵𝑝−𝜆_∞𝑘=1𝜙_𝑘𝑝,𝛼;𝐵,𝑎,𝑐𝑎_𝑘+𝑝−(𝐴−𝐵)≤0, (2.5) this implies 𝑓(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼).
For necessary condition, let 𝑓(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼) be given by (1.1), then 𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′𝑧^𝑝−1=𝑝−^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘𝑧^𝑘, (2.6) from (1.13) and (2.6), we find that 𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑧^𝑝−1−𝛼𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑝𝑧^𝑝−1−1−𝑝𝐵𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑧^𝑝−1−𝛼𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)^′/𝑝𝑧^𝑝−1−(=−∑−1𝑝𝐵+𝐴−𝐵)𝑝−𝜆_∞𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−(𝐴−𝐵)𝑝−𝜆−𝐵∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼𝐵−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘<1. (2.7)
Now, since |Re𝑧|≤|𝑧| for all 𝑧, we have Re−∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−(𝐴−𝐵)𝑝−𝜆−𝐵∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼𝐵−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘<1. (2.8) We choose value of 𝑧 on the real axis so that the expression (𝐿_𝑝(𝑎,𝑐)𝑓(𝑧))^′/𝑧^𝑝−1 is real.
Then, we have Re−∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−(𝐴−𝐵)𝑝−𝜆−𝐵∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼𝐵−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘=−∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−(𝐴−𝐵)𝑝−𝜆−𝐵∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼𝐵−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘,−∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−(𝐴−𝐵)𝑝−𝜆−𝐵∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼𝐵−∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘=−∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘|𝑧|^𝑘−(𝐴−𝐵)𝑝−𝜆−𝐵∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘𝑧^𝑘−𝛼𝐵∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘|𝑧|^𝑘<1. (2.9)(2.10)
Letting 𝑧→1⁻ throughout real values in (2.10), we get −∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘−𝛼∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘−(𝐴−𝐵)𝑝−𝜆−𝐵∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘−𝛼𝐵∑^∞_𝑘=1𝑘+𝑝/𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘<1. (2.11) So we have ∑^∞_𝑘=1𝑝𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘+𝛼∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘(𝐴−𝐵)𝑝−𝜆+𝐵∑^∞_𝑘=1𝑝𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘+𝛼𝐵∑^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘/(𝑐)_𝑘<1, (2.12) it is 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝𝑎_𝑘+𝑝(𝑎)_𝑘(𝑐)_𝑘≤𝑝(𝐴−𝐵)𝑝−𝜆. (2.13) This completes the proof of the theorem.

Corollary 2.2. Let the function 𝑓(𝑧)∈𝐴⁺(𝑝) be given by (1.1). If 𝑓(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼),then 𝑎_𝑘+𝑝≤𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑝+𝛼(1−𝐵)𝑘+𝑝(𝑎)_𝑘𝑘,𝑝∈𝑁. (2.14) The result is sharp for the function given by 𝑓(𝑧)=𝑧^𝑝−^∞_𝑘=1𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑝+𝛼(1−𝐵)𝑘+𝑝(𝑎)_𝑘𝑧^𝑘+𝑝. (2.15)

3. Distortion Inequality of Class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼)

Theorem 3.1. If a function 𝑓(𝑧) defined by (1.1) is in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), then 𝑝!!−()𝑝−𝑚𝑐𝑝𝐴−𝐵𝑝−𝜆𝑝!𝑎(!|𝑧||𝑧|𝑝+𝛼1−𝐵)𝑝+1−𝑚^𝑝−𝑚≤𝑓^(𝑚)(𝑧≤)𝑝!!+()𝑝−𝑚𝑐𝑝𝐴−𝐵𝑝−𝜆𝑝!𝑎(!|𝑧||𝑧|𝑝+𝛼1−𝐵)𝑝+1−𝑚^𝑝−𝑚𝑧∈𝑈;𝑎≥𝑐>0;𝑚∈𝑁₀;𝑝>𝑚. (3.1) The result is sharp for the function 𝑓(𝑧) given by 𝑓(𝑧)=𝑧^𝑝−𝑐(𝐴−𝐵)𝑝−𝜆𝑎()𝑧1−𝐵𝑝+𝛼^1+𝑝𝑝∈𝑁. (3.2)

Proof. Since 𝑓(𝑧)=𝑧^𝑝−∑^∞_𝑘=1|𝑎_𝑘+𝑝|𝑧^𝑘+𝑝, then 𝑓^(𝑚)(𝑧)=𝑝!!𝑧𝑝−𝑚^𝑝−𝑚−^∞_𝑘=1𝑘+𝑝!!𝑎𝑘+𝑝−𝑚^𝑘+𝑝𝑧^{𝑘+𝑝−𝑚}. (3.3)
From 𝑓(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), using Theorem 2.1, we have 𝑎𝑝+𝛼(1−𝐵)1+𝑝𝑐𝑝(𝐴−𝐵)𝑝−𝜆𝑝+1!^∞_𝑘=1𝑘+𝑝!𝑎_𝑘+𝑝≤^∞_𝑘=1𝑝+𝛼𝑘+𝑝(1−𝐵)(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑎_𝑘+𝑝≤1 (3.4) which readily yields 𝑘+𝑝!^∞_𝑘=1𝑎_𝑘+𝑝≤𝑐𝑝(𝐴−𝐵)𝑝−𝜆𝑝!𝑎𝑝+𝛼(1−𝐵). (3.5) By (3.3) and (3.5), we can imply (3.1).

4. Radius of Starlikeness and Convexity

Theorem 4.1. Let the function 𝑓(𝑧) defined by (1.1) be in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼). Then, 𝑓(𝑧) is starlike of 𝜂(0≤𝜂<1) in |𝑧|<𝑟_𝜂(𝑝,𝑘,𝛼,𝐴,𝐵,𝑎,𝑐) where 𝑟_𝜂𝑝𝜙𝑝,𝑘,𝛼,𝐴,𝐵,𝑎,𝑐=inf1−𝜂_𝑘𝑝,𝛼;𝐵,𝑎,𝑐(𝐴−𝐵)𝑘+𝑝1−𝜂^1/𝑘 (4.1) and 𝜙_𝑘(𝑝,𝛼;𝐵,𝑎,𝑐) is defined by (2.1).

Proof. We must show that 𝑧𝑓^′(𝑧)(𝑧)|𝑧|𝑝𝑓−1<1−𝜂for<𝑟_𝜂𝑝,𝑘,𝛼,𝐴,𝐵,𝑎,𝑐. (4.2) Since 𝑧𝑓^′(𝑧)(𝑧)=𝑝𝑓−1−∑^∞_𝑘=1𝑘𝑎_𝑘+𝑝𝑧^𝑘𝑝−∑^∞_𝑘=1𝑝𝑎_𝑘+𝑝𝑧^𝑘≤∑^∞_𝑘=1𝑘𝑎_𝑘+𝑝|𝑧|^𝑘𝑝−∑^∞_𝑘=1𝑝𝑎_𝑘+𝑝|𝑧|^𝑘, (4.3) to prove (4.1), it is sufficient to prove ∑^∞_𝑘=1𝑘𝑎_𝑘+𝑝|𝑧|^𝑘𝑝−∑^∞_𝑘=1𝑝𝑎_𝑘+𝑝|𝑧|^𝑘<1−𝜂. (4.4) It is equivalent to 𝑘+𝑝1−𝜂𝑎^∞_{𝑘=1𝑘+𝑝}𝑝|𝑧|1−𝜂^𝑘≤1. (4.5)
By Theorem 2.1, we have ^∞_𝑘=1𝜙_𝑘𝑝,𝛼;𝐵,𝑎,𝑐𝑎𝐴−𝐵^𝑘+𝑝≤1, (4.6) hence (4.5) will be true if 𝑘+𝑝1−𝜂𝑎_𝑘+𝑝𝑝|𝑧|1−𝜂^𝑘≤𝜙_𝑘𝑝,𝛼;𝐵,𝑎,𝑐𝑎𝐴−𝐵^𝑘+𝑝. (4.7) It is equivalent to |𝑧|≤𝑝1−𝜂𝜙_𝑘𝑝,𝛼;𝐵,𝑎,𝑐(𝐴−𝐵)𝑘+𝑝1−𝜂^1/𝑘. (4.8) This completes the proof.

Theorem 4.2. Let the function 𝑓(𝑧) defined by (1.1) be in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼). Then, 𝑓(𝑧) is convex of 𝜉(0≤𝜉<1) in |𝑧|<𝑟_𝜉(𝑝,𝑘,𝛼,𝐴,𝐵,𝑎,𝑐) where 𝑟_𝜉𝑝𝑝,𝑘,𝛼,𝐴,𝐵,𝑎,𝑐=inf²(1−𝜉)𝜙_𝑘𝑝,𝛼;𝐵,𝑎,𝑐𝑘+𝑝(𝐴−𝐵)𝑘+𝑝(1−𝜉)^1/𝑘 (4.9) and 𝜙_𝑘(𝑝,𝛼;𝐵,𝑎,𝑐) is defined by (2.1).

Proof. It sufficient to show that 𝑧𝑓^′′(𝑧)𝑝𝑓^′(𝑧)−𝑝−1𝑝<1−𝜉for|𝑧|<𝑟_𝜉𝑝,𝑘,𝛼,𝐴,𝐵,𝑎,𝑐. (4.10) Since 𝑧𝑓^′′(𝑧)𝑝𝑓^′(𝑧)−𝑝−1𝑝=−∑^∞_𝑘=1𝑘𝑘+𝑝𝑎_𝑘+𝑝𝑧^𝑘𝑝²−∑^∞_𝑘=1𝑝𝑘+𝑝𝑎_𝑘+𝑝𝑧^𝑘≤∑^∞_𝑘=1𝑘𝑘+𝑝𝑎_𝑘+𝑝|𝑧|^𝑘𝑝²−∑^∞_𝑘=1𝑝𝑘+𝑝𝑎_𝑘+𝑝|𝑧|^𝑘, (4.11) to prove (4.9), it is sufficient to prove ∑^∞_𝑘=1𝑘𝑘+𝑝𝑎_𝑘+𝑝|𝑧|^𝑘𝑝²−∑^∞_𝑘=1𝑝𝑘+𝑝𝑎_𝑘+𝑝|𝑧|^𝑘<1−𝜉. (4.12) It is equivalent to 𝑘+𝑝𝑘+𝑝(^∞_𝑘=11−𝜉)𝑎_𝑘+𝑝𝑝²()|𝑧|1−𝜉^𝑘≤1. (4.13)
By Theorem 2.1, we have ^∞_𝑘=1𝜙_𝑘𝑝,𝛼;𝐵,𝑎,𝑐𝑎𝐴−𝐵^𝑘+𝑝≤1, (4.14) hence (4.13) will be true if 𝑘+𝑝𝑘+𝑝(1−𝜉)𝑎_𝑘+𝑝𝑝²()|𝑧|1−𝜉^𝑘≤𝜙_𝑘𝑝,𝛼;𝐵,𝑎,𝑐𝑎𝐴−𝐵^𝑘+𝑝. (4.15) It is equivalent to |𝑧|≤𝑝²(1−𝜉)𝜙_𝑘𝑝,𝛼;𝐵,𝑎,𝑐𝑘+𝑝(𝐴−𝐵)𝑘+𝑝(1−𝜉)^1/𝑘. (4.16) This completes the proof.

5. δ-Neighborhood of 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼)

Based on the earlier works by Aouf et al. [6], Altintas et al. [13–15], and Aouf [16], we introduce the 𝛿-neighborhood of a function 𝑓(𝑧)∈𝐴⁺(𝑝) of the form (1.1) and present the relationship between 𝛿-neighborhood and corresponding function class.

Definition 5.1. For 𝛿>0,𝑎>0,𝑐>0, the 𝛿-neighborhood of a function 𝑓(𝑧)∈𝐴⁺(𝑝) is defined as follows: 𝑁⁺_𝛿𝑓=𝑔∶𝑔(𝑧)=𝑧^𝑝−^∞_𝑘=1𝑏_𝑘+𝑝𝑧^𝑘+𝑝∈𝐴𝑝,^∞_𝑘=1𝑡_𝑘𝑏_𝑘+𝑝−𝑎_𝑘+𝑝<𝛿, (5.1) where 𝑡_𝑘=𝑝+𝛼(1−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘. (5.2)

Theorem 5.2. Let the function 𝑓(𝑧) defined by (1.1) be in the class 𝑌⁺_𝑎+1,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼). Then, 𝑁⁺_𝛿𝑓⊂𝑌⁺_𝑎,𝑐1𝐴,𝐵;𝑝,𝜆,𝛼𝑎+1. (5.3) This result is the best possible in the sense that 𝛿 cannot be increased.

Proof. Let 𝑓(𝑧)∈𝑌⁺_𝑎+1,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), then by Theorem 2.1, we have 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝(𝑎+1)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑎_𝑘+𝑝≤1 (5.4) which is equivalent to 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑎_𝑘+𝑝≤𝑎𝑎+1. (5.5)
For any 𝑔(𝑧)=𝑧^𝑝−^∞_𝑘=1𝑏_𝑘+𝑝𝑧^𝑘+𝑝∈𝑁⁺_𝛿𝑓, (5.6) we find from (5.1) that 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑏_𝑘+𝑝−𝑎_𝑘+𝑝≤𝛿. (5.7) By (5.5) and (5.7), we get 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑏_𝑘+𝑝≤^∞_𝑘=1𝑝+𝛼(1−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑎_𝑘+𝑝+^∞_𝑘=1𝑝+𝛼(1−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑏_𝑘+𝑝−𝑎_𝑘+𝑝≤𝑎𝑎+1+𝛿=1. (5.8) By Theorem 2.1, it implies that 𝑔(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼).
To show the sharpness of the assertion of Theorem 5.2, we consider the functions 𝑓(𝑧) and 𝑔(𝑧) given by 𝑓(𝑧)=𝑧^𝑝−𝑐𝑝(𝐴−𝐵)𝑝−𝜆()()𝑧𝑎+1𝑝+𝛼1−𝐵𝑝+1^𝑝+1∈𝑌⁺_𝑎+1,𝑐,𝑔(𝑧)𝐴,𝐵;𝑝,𝜆,𝛼=𝑧^𝑝−𝑐𝑝(𝐴−𝐵)𝑝−𝜆()()+()𝑎+1𝑝+𝛼1−𝐵𝑝+1𝑐𝑝𝐴−𝐵𝑝−𝜆𝛿′𝑎()𝑧𝑝+𝛼1−𝐵1+𝑝^𝑝+1, (5.9) where 𝛿′>𝛿=1/(𝑎+1).
Since 𝑝+𝛼𝑝+1(1−𝐵)(𝑎)₁𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)₁𝑏_1+𝑝−𝑎_1+𝑝=𝑝+𝛼𝑝+1(1−𝐵)(𝑎)₁𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)₁⋅𝑐𝑝(𝐴−𝐵)𝑝−𝜆𝛿′𝑎𝑝+𝛼(1−𝐵)1+𝑝=𝛿^′, (5.10) so 𝑔(𝑧)∈𝑁⁺_𝛿′(𝑓).
But 𝑝+𝛼𝑝+1(1−𝐵)(𝑎)₁𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)₁𝑐𝑝(𝐴−𝐵)𝑝−𝜆(()+()𝛿𝑎+1)𝑝+𝛼𝑝+11−𝐵𝑐𝑝𝐴−𝐵𝑝−𝜆^′𝑎(=𝑎𝑝+𝛼1−𝐵)1+𝑝𝑎+1+𝛿^′>1, (5.11) by Theorem 2.1𝑔(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼).

6. Properties Associated with Modified Hadamard Product

Following early works by Aouf et al. in [6], we provide the properties of modified Hadamard product of 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼).

For the function 𝑓_𝑗(𝑧)=𝑧^𝑝−^∞_𝑘=1𝑎_{𝑘+𝑝,𝑗}𝑧^𝑘+𝑝𝑗=1,2;𝑝∈𝑁, (6.1) the modified Hadamard product of the functions 𝑓₁(𝑧) and 𝑓₂(𝑧) was denoted by (𝑓₁⋅𝑓₂)(𝑧) and defined as follows: 𝑓₁⋅𝑓₂(𝑧)=𝑧^𝑝−^∞_𝑘=1𝑎_𝑘+𝑝,1𝑎_𝑘+𝑝,2𝑧^𝑘+𝑝=𝑓₂⋅𝑓₁(𝑧). (6.2)

Theorem 6.1. Let 𝑓_𝑗(𝑧)(𝑗=1,2) given by (6.1) be in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), then 𝑓₁⋅𝑓₂(𝑧)∈𝑌⁺_𝑎,𝑐𝐴,𝐵;𝑝,𝛾,𝛼, (6.3) where (𝛾∶=𝑝−𝑐𝑝𝐴−𝐵)𝑝−𝜆²𝑎𝑝+𝛼(1−𝐵)1+𝑝. (6.4) The result is sharp for the functions 𝑓_𝑗(𝑧)(𝑗=1,2) given by 𝑓_𝑗(𝑧)=𝑧^𝑝−𝑐𝑝(𝐴−𝐵)𝑝−𝜆𝑎()𝑧𝑝+𝛼1−𝐵1+𝑝^𝑝+1. (6.5)

Proof. By Theorem 2.1, we need to find the largest 𝛾 such that 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝛾(𝑐)_𝑘𝑎_𝑘+𝑝,1⋅𝑎_𝑘+𝑝,2≤1. (6.6)
Since 𝑓_𝑗(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼),(𝑗=1,2), then we see that 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑎_{𝑘+𝑝,𝑗}≤1𝑗=1,2. (6.7)
By Cauchy-Schwartz inequality, we obtain 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑎_𝑘+𝑝,1⋅𝑎_𝑘+𝑝,2≤1. (6.8)
This implies that we only need to show that 𝑎_𝑘+𝑝,1⋅𝑎_𝑘+𝑝,2≤𝑝−𝛾𝑎_𝑘+𝑝,1⋅𝑎_𝑘+𝑝,2()𝑝−𝜆𝑘∈𝑁 (6.9) or equivalently that 𝑎_𝑘+𝑝,1⋅𝑎_𝑘+𝑝,2≤𝑝−𝛾𝑝−𝜆. (6.10)
By making use of the inequality (6.8), it is sufficient to prove that 𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑝+𝛼(1−𝐵)𝑘+𝑝(𝑎)_𝑘≤𝑝−𝛾()𝑝−𝜆𝑘∈𝑁. (6.11) From (6.11), we have 𝑝(𝛾≤𝑝−𝐴−𝐵)𝑝−𝜆²(𝑐)_𝑘𝑝+𝛼(1−𝐵)𝑘+𝑝(𝑎)_𝑘(𝑘∈𝑁). (6.12)
Define the function Φ(𝑘) by Φ(𝑘)𝑝(∶=𝑝−𝐴−𝐵)𝑝−𝜆²(𝑐)_𝑘𝑝+𝛼(1−𝐵)𝑘+𝑝(𝑎)_𝑘(𝑘∈𝑁). (6.13) We see that Φ(𝑘) is an increasing function of 𝑘. Therefore, we conclude that (1)𝑝(𝛾≤Φ=𝑝−𝐴−𝐵)𝑝−𝜆²(𝑐)₁𝑝+𝛼(1−𝐵)𝑘+𝑝(𝑎)₁. (6.14)
By using arguments similar to those in the proof of Theorem 6.1, we can derive the following result.

Theorem 6.2. Let 𝑓₁(𝑧) defined by (6.1) be in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), 𝑓₂(𝑧) defined by (6.1) be in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝛾,𝛼), then 𝑓₁⋅𝑓₂(𝑧)∈𝑌⁺_𝑎,𝑐𝐴,𝐵;𝑝,𝜉,𝛼, (6.15) where ()𝜉∶=𝑝−𝑐𝑝𝐴−𝐵𝑝−𝜆𝑝−𝛾𝑎𝑝+𝛼(1−𝐵)1+𝑝. (6.16) The result is sharp for the functions 𝑓_𝑗(𝑧)(𝑗=1,2) given by 𝑓₁(𝑧)=𝑧^𝑝−𝑐𝑝(𝐴−𝐵)𝑝−𝜆𝑎()𝑧𝑝+𝛼1−𝐵1+𝑝^𝑝+1,𝑓𝑝∈𝑁₂(𝑧)=𝑧^𝑝−𝑐𝑝(𝐴−𝐵)𝑝−𝛾𝑎(𝑧𝑝+𝛼1−𝐵)1+𝑝^𝑝+1𝑝∈𝑁. (6.17)

Theorem 6.3. Let 𝑓_𝑗(𝑧)(𝑗=1,2) defined by (6.1) be in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), then the function ℎ(𝑧) defined by ℎ(𝑧)=𝑧^𝑝−^∞_𝑘=1𝑎_𝑘+𝑝,1²+𝑎_𝑘+𝑝,2² (6.18) belongs to the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜒,𝛼) where ()𝜒∶=𝑝−2𝑝𝑐𝐴−𝐵𝑝−𝜆𝑎𝑝+𝛼(1−𝐵)1+𝑝. (6.19) This result is sharp for the functions given by (6.5).

Proof. By Theorem 2.1, we want to find the largest 𝜒 such that 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜒(𝑐)_𝑘𝑎_𝑘+𝑝,1²+𝑎_𝑘+𝑝,2²≤1. (6.20)
Since 𝑓_𝑗(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼)(𝑗=1,2), we readily see that 𝑝+𝛼(^∞_𝑘=11−𝐵)𝑘+𝑝(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜒(𝑐)_𝑘𝑎_{𝑘+𝑝,𝑗}≤1𝑗=1,2. (6.21)
From (6.21), we have ^∞_𝑘=1𝑝+𝛼²(1−𝐵)²𝑘+𝑝²𝑝²(𝐴−𝐵)²𝑝−𝜒²(𝑎)_𝑘(𝑐)_𝑘²𝑎_{𝑘+𝑝,𝑗}²≤1𝑗=1,2, (6.22) then we have ^∞_𝑘=1𝑝+𝛼²(1−𝐵)²𝑘+𝑝²𝑝²(𝐴−𝐵)²𝑝−𝜒²(𝑎)_𝑘(𝑐)_𝑘²𝑎_𝑘+𝑝,1²+𝑎_𝑘+𝑝,2²≤2. (6.23)
From (6.23), if we want to prove (6.20), it is sufficient to prove there exists the largest 𝜒 such that 1≤(𝑎)𝑝−𝜒𝑝+𝛼𝑘+𝑝1−𝐵)(_𝑘2𝑝(𝐴−𝐵)𝑝−𝜆²(𝑐)_𝑘(𝑘∈𝑁), (6.24) that is (𝜒≤𝑝−2𝑝𝐴−𝐵)𝑝−𝜆²(𝑐)_𝑘𝑝+𝛼𝑘+𝑝(1−𝐵)(𝑎)_𝑘(𝑘∈𝑁). (6.25)
Now we define Ψ(𝑘) by Ψ(𝑘)(=𝑝−2𝑝𝐴−𝐵)𝑝−𝜆²(𝑐)_𝑘𝑝+𝛼𝑘+𝑝(1−𝐵)(𝑎)_𝑘(𝑘∈𝑁). (6.26) We see that Ψ(𝑘) is an increasing function of 𝑘. Therefore, we conclude that (1)()𝜒≤Ψ=𝑝−2𝑝𝑐𝐴−𝐵(𝑝−𝜆)²𝑎𝑝+𝛼𝑘+𝑝(1−𝐵) (6.27) which completes the proof of Theorem 6.1.

7. Application of Fractional Calculus Operator

References [17–19] have studied the fractional calculus operators extensively. In this part, we only investigate the application of fractional calculus operator which was defined by [6] in the class of 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼).

Definition 7.1 (see [6]). The fractional integral of order 𝜇 is defined, for a function 𝑓(𝑧), by 𝐷_𝑧^−𝜇𝑓(𝑧)=1Γ𝜇^𝑧₀𝑓(𝑧)(𝑧−𝜉)^1−𝜇𝑑𝜉𝜇>0, (7.1) where the function 𝑓(𝑧) is analytic in a simply connected domain of the complex z-plane containing the origin and the multiplicity of (𝑧−𝜉)^𝜇−1 is removed by requiring log(𝑧−𝜉) to be real when 𝑧−𝜉>0.

Definition 7.2 (see [6]). The fractional integral of order 𝜇 is defined, for a function 𝑓(𝑧), by 𝐷^𝜇_𝑧𝑓(𝑧)=1Γ1−𝜇^𝑧₀𝑓(𝑧)(𝑧−𝜉)^𝜇𝑑𝜉0≤𝜇<1, (7.2) where the function 𝑓(𝑧) is constrained, the multiplicity of (𝑧−𝜉)^−𝜇 is removed as in Definition 7.1.

In our investigation, we will use the operators 𝐽_𝛿,𝑝 defined by (cf, [20–22]) 𝐽_𝛿,𝑝𝑓(𝑧)∶=𝛿+𝑝𝑧^𝑝𝑧₀𝑡^𝛿−1𝑓(𝑡)𝑑𝑡𝑓∈𝐴𝑝;𝛿>−𝑝;𝑝∈𝑁 (7.3) as well as 𝐷^𝜇_𝑧 for which it is well known that (see [23]) 𝐷^𝜇_𝑧𝑧^𝜌=Γ𝜌+1Γ𝑧𝜌−𝜇+1^𝜌−𝜇𝜌>−1;𝜇∈𝑅 (7.4) in terms of Gamma.

Lemma 7.3 (see Chen et al. [18]). For a function 𝑓(𝑧)∈𝐴(𝑝), 𝐷^𝜇_𝑧𝐽_𝛿,𝑝𝑓(𝑧=Γ)𝑝+1Γ𝑧𝑝−𝜇+1^𝑝−𝜇+^∞_𝑘=1𝛿+𝑝Γ𝑘+𝑝+1Γ𝑎𝛿+𝑘+𝑝𝑘+𝑝−𝜇+1^𝑘+𝑝𝑧^{𝑘+𝑝−𝜇},𝐽_𝛿,𝑝𝐷^𝜇_𝑧𝑓(𝑧=Γ)𝛿+𝑝𝑝+1Γ𝑧𝛿+𝑝−𝜇𝑝−𝜇+1^𝑝−𝜇+^∞_𝑘=1𝛿+𝑝Γ𝑘+𝑝+1Γ𝑎𝛿+𝑘+𝑝−𝜇𝑘+𝑝−𝜇+1^𝑘+𝑝𝑧^{𝑘+𝑝−𝜇} (7.5)(𝜇∈𝑅;𝛿>−𝑝;𝑝∈𝑁) provided that no zeros appear in the denominators in (7.5).

Remark 7.4. Throughout this section, we assume further that 𝑎≥𝑐>0.

Theorem 7.5. Let function defined by (1.1) be in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), then 𝐷_𝑧^−𝜇𝐽_𝛿,𝑝𝑓(𝑧≥Γ)𝑝+1Γ|𝑧|𝑝+𝜇+1^𝑝+𝜇)1−𝑐𝑝𝛿+𝑝(𝐴−𝐵𝑝−𝜆𝑎()|𝑧|𝐷𝑝+𝛼𝑝+𝛿+1𝑝+𝜇+11−𝐵_𝑧−𝜇,𝐽_𝛿,𝑝𝑓(𝑧≤Γ)𝑝+1Γ|𝑧|𝑝+𝜇+1^𝑝+𝜇)1+𝑐𝑝𝛿+𝑝(𝐴−𝐵𝑝−𝜆𝑎()|𝑧|𝑝+𝛼𝑝+𝛿+1𝑝+𝜇+11−𝐵 (7.6)(7.7)𝜇>0;𝛿>−𝑝;𝑝∈𝑁;𝑧∈𝑈, each of the assertions is sharp.

Proof. Since 𝐷_𝑧^−𝜇𝐽_𝛿,𝑝𝑓(𝑧=Γ)𝑝+1Γ𝑧𝑝−𝜇+1^𝑝+𝜇−^∞_𝑘=1𝛿+𝑝Γ𝑘+𝑝+1Γ𝑎𝛿+𝑘+𝑝𝑘+𝑝+𝜇+1^𝑘+𝑝𝑧^{𝑘+𝑝+𝜇}, (7.8) then 𝐷_𝑧^−𝜇𝐽_𝛿,𝑝𝑓(𝑧=Γ)𝑝+1Γ𝑧𝑝−𝜇+1^𝜇𝑧^𝑝−^∞_𝑘=1𝛿+𝑝Γ𝑘+𝑝+1Γ𝑝+𝜇+1ΓΓ𝑎𝛿+𝑘+𝑝𝑘+𝑝+𝜇+1𝑝+1^𝑘+𝑝𝑧^𝑘+𝑝. (7.9)
Let 𝐺(𝑧) is defined by 𝐺(𝑧)=𝑧^𝑝−^∞_𝑘=1𝛿+𝑝Γ𝑘+𝑝+1Γ𝑝+𝜇+1ΓΓ𝑎𝛿+𝑘+𝑝𝑘+𝑝+𝜇+1𝑝+1^𝑘+𝑝𝑧^𝑘+𝑝=𝑧^𝑝−^∞_𝑘=1𝑄(𝑘)𝑎_𝑘+𝑝𝑧^𝑘+𝑝, (7.10) where 𝑄(𝑘)=𝛿+𝑝Γ𝑘+𝑝+1Γ𝑝+𝜇+1𝛿+𝑘+𝑝Γ𝑘+𝑝+𝜇+1Γ𝑝+1𝑘,𝑝∈𝑁;𝜇>0. (7.11)
Since 𝑄(𝑘) is a decreasing function of 𝑘 when 𝜇>0, we get 0<𝑄(𝑘)≤𝑄(1)=𝛿+𝑝𝑝+1𝛿+𝑘+𝑝1+𝑝+𝜇, (7.12) since 𝑓(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), by Theorem 2.1, we get 𝑎𝑝+𝛼(1−𝐵)𝑝+1𝑐𝑝(𝐴−𝐵)𝑝−𝜆^∞_𝑘=1𝑎_𝑘+𝑝≤^∞_𝑘=1𝑝+𝛼𝑘+𝑝(1−𝐵)(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑎_𝑘+𝑝≤1. (7.13)
It is that ^∞_𝑘=1𝑎_𝑘+𝑝≤𝑐𝑝(𝐴−𝐵)𝑝−𝜆𝑎𝛼+𝑝(1−𝐵)𝑝+1, (7.14) then |𝐺(𝑧)|=𝑧^𝑝−^∞_𝑘=1𝑄(𝑘)𝑎_𝑘+𝑝𝑧^𝑘+𝑝≥|𝑧|^𝑝−𝑄(1)|𝑧|^∞𝑝+1_𝑘=1𝑎_𝑘+𝑝≥|𝑧|^𝑝−𝑐𝑝𝑝+𝛿(𝐴−𝐵)𝑝−𝜆𝑎()|𝑧|𝑝+𝛼𝛿+𝑘+𝑝𝑝+1+𝜇1−𝐵^𝑝+1,|𝐺(𝑧)|=𝑧^𝑝−^∞_𝑘=1𝑄(𝑘)𝑎_𝑘+𝑝𝑧^𝑘+𝑝≤|𝑧|^𝑝+𝑄(1)|𝑧|^∞𝑝+1_𝑘=1𝑎_𝑘+𝑝≤|𝑧|^𝑝+𝑐𝑝𝑝+𝛿(𝐴−𝐵)𝑝−𝜆𝑎()|𝑧|𝑝+𝛼𝛿+𝑘+𝑝𝑝+1+𝜇1−𝐵^𝑝+1. (7.15) From (7.15), we obtain (7.6) and (7.7), respectively.
Equalities in (7.6) and (7.7) are attained for the function 𝑓(𝑧) given by 𝐷_𝑧^−𝜇𝐽_𝛿,𝑝𝑓(𝑧=Γ)𝑝+1Γ𝑧𝑝−𝜇+1^𝑝+𝜇)1−𝑐𝑝𝛿+𝑝(𝐴−𝐵𝑝−𝜆𝑎()𝑧𝑝+𝛼𝛿+𝑘+𝑝𝑝+𝜇+11−𝐵. (7.16)

Theorem 7.6. Let function defined by (1.1) be in the class 𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), then 𝐷^𝜇_𝑧𝐽_𝛿,𝑝𝑓(𝑧≥Γ)𝑝+1Γ|𝑧|𝑝−𝜇+1^𝑝−𝜇)1−𝑐𝑝𝛿+𝑝(𝐴−𝐵𝑝−𝜆𝑎()|𝑧|𝐷𝑝+𝛼𝑝+𝛿+1𝑝−𝜇+11−𝐵_𝜇𝑧,𝐽_𝛿,𝑝𝑓(𝑧≤Γ)𝑝+1Γ|𝑧|𝑝−𝜇+1^𝑝−𝜇)1+𝑐𝑝𝛿+𝑝(𝐴−𝐵𝑝−𝜆𝑎()|𝑧|𝑝+𝛼𝑝+𝛿+1𝑝−𝜇+11−𝐵, (7.17)(7.18)𝜇>0;𝛿>−𝑝;𝑝∈𝑁;𝑧∈𝑈, each of the assertions is sharp.

Proof. Since 𝐷^𝜇_𝑧𝐽_𝛿,𝑝𝑓(𝑧=Γ)𝑝+1Γ𝑧𝑝−𝜇+1^𝑝−𝜇−^∞_𝑘=1𝛿+𝑝Γ𝑘+𝑝+1Γ𝑎𝛿+𝑘+𝑝𝑘+𝑝−𝜇+1^𝑘+𝑝𝑧^{𝑘+𝑝−𝜇}, (7.19) then 𝐷^𝜇_𝑧𝐽_𝛿,𝑝𝑓(𝑧=Γ)𝑝+1Γ𝑧𝑝−𝜇+1^−𝜇𝑧^𝑝−^∞_𝑘=1𝛿+𝑝Γ𝑘+𝑝+1Γ𝑝−𝜇+1ΓΓ𝑎𝛿+𝑘+𝑝𝑘+𝑝−𝜇+1𝑝+1^𝑘+𝑝𝑧^𝑘+𝑝. (7.20)
Let 𝐻(𝑧) be defined by 𝐻(𝑧)=𝑧^𝑝−^∞_𝑘=1𝛿+𝑝Γ𝑘+𝑝+1Γ𝑝−𝜇+1ΓΓ𝑎𝛿+𝑘+𝑝𝑘+𝑝−𝜇+1𝑝+1^𝑘+𝑝𝑧^𝑘+𝑝=𝑧^𝑝−^∞_𝑘=1𝐸(𝑘)𝑘+𝑝𝑎_𝑘+𝑝𝑧^𝑘+𝑝, (7.21) where 𝐸(𝑘)=𝛿+𝑝Γ𝑘+𝑝Γ𝑝−𝜇+1𝛿+𝑘+𝑝Γ𝑘+𝑝−𝜇+1Γ𝑝+1𝑘,𝑝∈𝑁;𝜇>0. (7.22)
Since 𝐸(𝑘) is a decreasing function of 𝑘 when 0≤𝜇<1, then 0<𝐸(𝑘)≤𝐸(1)=𝛿+𝑝𝛿+1+𝑝1+𝑝−𝜇, (7.23) since 𝑓(𝑧)∈𝑌⁺_𝑎,𝑐(𝐴,𝐵;𝑝,𝜆,𝛼), by Theorem 2.1, we get 𝑎𝑝+𝛼(1−𝐵)𝑐𝑝(𝐴−𝐵)𝑝−𝜆^∞_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝≤^∞_𝑘=1𝑝+𝛼𝑘+𝑝(1−𝐵)(𝑎)_𝑘𝑝(𝐴−𝐵)𝑝−𝜆(𝑐)_𝑘𝑎_𝑘+𝑝≤1. (7.24)
It is that 𝑘+𝑝𝑎^∞_{𝑘=1𝑘+𝑝}≤𝑐𝑝(𝐴−𝐵)𝑝−𝜆𝑎𝛼+𝑝(1−𝐵), (7.25) then |𝐻(𝑧)|=𝑧^𝑝−^∞_𝑘=1𝐸(𝑘)𝑘+𝑝𝑎_𝑘+𝑝𝑧^𝑘+𝑝≥|𝑧|^𝑝−𝐸(1)|𝑧|^∞𝑝+1_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝≥|𝑧|^𝑝−𝑐𝑝𝑝+𝛿(𝐴−𝐵)𝑝−𝜆𝑎()|𝑧|𝑝+𝛼𝛿+𝑘+𝑝𝑝+1−𝜇1−𝐵^𝑝+1,|𝐻(𝑧)|=𝑧^𝑝−^∞_𝑘=1𝐸(𝑘)𝑘+𝑝𝑎_𝑘+𝑝𝑧^𝑘+𝑝≤|𝑧|^𝑝+𝐸(1)|𝑧|^∞𝑝+1_𝑘=1𝑘+𝑝𝑎_𝑘+𝑝≤|𝑧|^𝑝+𝑐𝑝𝑝+𝛿(𝐴−𝐵)𝑝−𝜆𝑎()|𝑧|𝑝+𝛼𝛿+𝑘+𝑝𝑝+1−𝜇1−𝐵^𝑝+1. (7.26) From (7.26), we obtain (7.17) and (7.18), respectively.
Equalities in (7.17) and (7.18) are attained for the function 𝑓(𝑧) given by 𝐷^𝜇_𝑧𝐽_𝛿,𝑝𝑓(𝑧=Γ)𝑝+1Γ𝑧𝑝−𝜇+1^𝑝−𝜇)1−𝑐𝑝𝛿+𝑝(𝐴−𝐵𝑝−𝜆𝑎()𝑧𝑝+𝛼𝛿+𝑘+𝑝𝑝−𝜇+11−𝐵 (7.27) or equivalently by 𝐽_𝛿,𝑝𝑓(𝑧)=𝑧^𝑝−𝑐𝑝𝛿+𝑝(𝐴−𝐵)𝑝−𝜆𝑎()𝑧𝑝+𝛼𝛿+𝑘+𝑝𝑝−𝜇+11−𝐵^𝑝+1. (7.28)

Acknowledgments

The author thanks the referee for numerous suggestions that helped make this paper more readable. The present investigation was partly supported by the Natural Science Foundation of Inner Mongolia under Grant 2009MS0113.