Convexity of Certain q Integral Operators of p Valent Functions

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Research Article

Convexity of Certain

𝑞-Integral Operators of 𝑝-Valent Functions

K. A. Selvakumaran,

1

S. D. Purohit,

2

Aydin Secer,

3

and Mustafa Bayram

3

1_{Department of Mathematics, RMK College of Engineering and Technology, Puduvoyal, Tamil Nadu 601206, India}

2_{Department of Basic Sciences (Mathematics), College of Technology and Engineering, M. P. University of Agriculture and Technology,}

Udaipur, Rajasthan 313001, India

3_{Department of Mathematical Engineering, Yildiz Technical University, Davutpasa, 34210 Istanbul, Turkey}

Correspondence should be addressed to Aydin Secer; asecer@yildiz.edu.tr Received 1 March 2014; Accepted 3 May 2014; Published 12 May 2014 Academic Editor: Guotao Wang

Copyright © 2014 K. A. Selvakumaran et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By applying the concept (and theory) of fractional𝑞-calculus, we first define and introduce two new 𝑞-integral operators for certain analytic functions defined in the unit discU. Convexity properties of these 𝑞-integral operators on some classes of analytic functions defined by a linear multiplier fractional𝑞-differintegral operator are studied. Special cases of the main results are also mentioned.

1. Introduction and Preliminaries

The subject of fractional calculus has gained noticeable importance and popularity due to its established applica-tions in many fields of science and engineering during the past three decades or so. Much of the theory of fractional calculus is based upon the familiar Riemann-Liouville frac-tional derivative (or integral). The fracfrac-tional 𝑞-calculus is the extension of the ordinary fractional calculus in the 𝑞-theory. Recently, there was a significant increase of activity in the area of the𝑞-calculus due to applications of the 𝑞-calculus in mathematics, statistics, and physics. For more details, one may refer to the books [1–4] on the subject. Recently, Purohit and Raina [5–7] have added one more dimension to this study by introducing certain subclasses of functions which are analytic in the open diskU, by using fractional 𝑞-calculus. Purohit [8] also studied similar work and considered new classes of multivalently analytic functions in the open unit disk.

The aim of this paper is to consider a linear multiplier fractional𝑞-differintegral operator and to define certain new subclasses of functions which are 𝑝-valent and analytic in the open unit disk. The results derived include convexity properties of these 𝑞-integral operators on some classes of analytic functions. Special cases of the main results are also mentioned.

LetA_𝑝denote the class of functions𝑓(𝑧) of the form 𝑓 (𝑧) = 𝑧𝑝+ ∑∞

𝑛=𝑝+1

𝑎_𝑛𝑧𝑛, (𝑝 ∈ N = {1, 2, 3, . . .}) , (1) which are analytic and𝑝-valent in the open unit disk U = {𝑧 ∈ C : |𝑧| < 1}. A function 𝑓 ∈ A_𝑝is said to be𝑝-valently starlike of order𝛼 (0 ≤ 𝛼 < 𝑝) if and only if

R {𝑧𝑓󸀠(𝑧)

𝑓 (𝑧) } > 𝛼, (𝑧 ∈ U) . (2) We denote byS∗_𝑝(𝛼) the class of all such functions. On the other hand, a function𝑓 ∈ A_𝑝is said to be in the classC_𝑝(𝛼) of𝑝-valently convex of order 𝛼 (0 ≤ 𝛼 < 𝑝) if and only if

R {1 +𝑧𝑓󸀠󸀠(𝑧)

𝑓󸀠_(𝑧) } > 𝛼, (𝑧 ∈ U) . (3)

Note thatS∗_𝑝(0) = S∗_𝑝andC_𝑝(0) = 𝐶_𝑝are, respectively, the classes of𝑝-valently starlike and 𝑝-valently convex functions inU. Also, we note that S∗₁(0) = S∗ andC₁(0) = C are, respectively, the usual classes of starlike and convex functions inU. A function 𝑓 ∈ A_𝑝is said to be in the classUS_𝑝(𝛼, 𝑘)

Volume 2014, Article ID 925902, 7 pages http://dx.doi.org/10.1155/2014/925902

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of𝑘-uniformly 𝑝-valent starlike of order 𝛼 (−1 ≤ 𝛼 < 𝑝) if it satisfies R {𝑧𝑓󸀠(𝑧) 𝑓 (𝑧) − 𝛼} ≥ 𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨 𝑧𝑓󸀠_(𝑧) 𝑓 (𝑧) − 𝑝󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨, (𝑘 ≥ 0, 𝑧 ∈ U) . (4) Furthermore, a function𝑓 ∈ A_𝑝 is said to be in the class UC_𝑝(𝛼, 𝑘) of 𝑘-uniformly 𝑝-valent convex of order 𝛼 (−1 ≤ 𝛼 < 𝑝) if it satisfies R {1 +𝑧𝑓󸀠󸀠(𝑧) 𝑓󸀠_(𝑧) − 𝛼} ≥ 𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨1 + 𝑧𝑓󸀠󸀠_(𝑧) 𝑓󸀠_(𝑧) − 𝑝󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨, (𝑘 ≥ 0, 𝑧 ∈ U) . (5)

For uniformly starlike and uniformly convex functions we refer to the papers [9–11]. Note thatUS₁(𝛼, 𝑘) = UST(𝛼, 𝑘) andUC₁(𝛼, 𝑘) = UCV(𝛼, 𝑘), where the classes UST(𝛼, 𝑘) andUCV(𝛼, 𝑘) are, respectively, the classes of 𝑘-uniformly starlike of order𝛼 (0 ≤ 𝛼 < 1) and 𝑘-uniformly convex of order𝛼 (0 ≤ 𝛼 < 1) studied in [12].

For the convenience of the reader, we now give some basic definitions and related details of𝑞-calculus which are used in the sequel.

For any complex number𝛼 the 𝑞-shifted factorials are defined as

(𝛼; 𝑞)₀= 1, (𝛼; 𝑞)_𝑛 =𝑛−1∏

𝑘=0

(1 − 𝛼𝑞𝑘) , 𝑛 ∈ N, (6) and in terms of the basic analogue of the gamma function

(𝑞𝛼; 𝑞)_𝑛= Γ𝑞(𝛼 + 𝑛) (1 − 𝑞)

𝑛

Γ_𝑞(𝛼) , (𝑛 > 0) , (7) where the𝑞-gamma function is defined by

Γ_𝑞(𝑥) = (𝑞, 𝑞)∞(1 − 𝑞)

1−𝑥

(𝑞𝑥_{; 𝑞)} ∞

, (0 < 𝑞 < 1) . (8) If|𝑞| < 1, the definition (6) remains meaningful for𝑛 = ∞ as a convergent infinite product:

(𝛼; 𝑞)_∞=∏∞

𝑗=0

(1 − 𝛼𝑞𝑗) . (9) In view of the relation

lim

𝑞 → 1−

(𝑞𝛼_{; 𝑞)} 𝑛

(1 − 𝑞)𝑛 = (𝛼)𝑛, (10) we observe that the 𝑞-shifted factorial (6) reduces to the familiar Pochhammer symbol (𝛼)_𝑛, where (𝛼)_𝑛 = 𝛼(𝛼 + 1) ⋅ ⋅ ⋅ (𝛼 + 𝑛 − 1). Also, the 𝑞-derivative and 𝑞-integral of a function on a subset ofC are, respectively, given by (see [2] pp. 19–22) 𝐷𝑞𝑓 (𝑧) = 𝑓 (𝑧) − 𝑓 (𝑧𝑞)_{(1 − 𝑞) 𝑧} , (𝑧 ̸= 0, 𝑞 ̸= 0) , ∫𝑧 0 𝑓 (𝑡) 𝑑𝑞𝑡 = 𝑧 (1 − 𝑞) ∞ ∑ 𝑘=0 𝑞𝑘𝑓 (𝑧𝑞𝑘) . (11)

Therefore, the𝑞-derivative of 𝑓(𝑧) = 𝑧𝑛, where𝑛 is a positive integer, is given by 𝐷_𝑞𝑧𝑛 =𝑧𝑛− (𝑧𝑞) 𝑛 (1 − 𝑞) 𝑧 = [𝑛]𝑞 𝑧𝑛−1, (12) where [𝑛]_𝑞= 1 − 𝑞𝑛 1 − 𝑞 = 𝑞𝑛−1+ ⋅ ⋅ ⋅ + 1 (13) and is called the𝑞-analogue of 𝑛. As 𝑞 → 1, we have [𝑛]_𝑞 = 𝑞𝑛−1_{+ ⋅ ⋅ ⋅ + 1 → 1 + ⋅ ⋅ ⋅ + 1 = 𝑛.}

The 𝑞-analogues to the function classes S∗_𝑝(𝛼), C_𝑝(𝛼), US_𝑝(𝛼, 𝑘), and UC_𝑝(𝛼, 𝑘) are given as follows.

A function𝑓 ∈ A_𝑝 is said to be in the classS∗_𝑞,𝑝(𝛼) of 𝑝-valently starlike with respect to 𝑞-differentiation of order 𝛼 (0 ≤ 𝛼 < 𝑝) if it satisfies

R {𝑧𝐷𝑞(𝑓 (𝑧))

𝑓 (𝑧) } > 𝛼, (𝑧 ∈ U) . (14) Also, a function𝑓 ∈ A_𝑝is said to be in the classC_𝑞,𝑝(𝛼) of 𝑝-valently convex with respect to 𝑞-differentiation of order 𝛼 (0 ≤ 𝛼 < 𝑝) if it satisfies

R {1 + 𝑧𝐷

2 𝑞(𝑓 (𝑧))

𝐷_𝑞(𝑓 (𝑧)) } > 𝛼, (𝑧 ∈ U) . (15) On the other hand, a function𝑓 ∈ A_𝑝is said to be in the class US_𝑞,𝑝(𝛼, 𝑘) of 𝑘-uniformly 𝑝-valent starlike with respect to 𝑞-differentiation of order 𝛼 (−1 ≤ 𝛼 < 𝑝) if it satisfies

R {𝑧𝐷𝑞(𝑓 (𝑧)) 𝑓 (𝑧) − 𝛼} ≥ 𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨 𝑧𝐷_𝑞(𝑓 (𝑧)) 𝑓 (𝑧) − [𝑝]𝑞󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨, (𝑘 ≥ 0, 𝑧 ∈ U) . (16)

Furthermore, a function𝑓 ∈ A_𝑝 is said to be in the class UC_𝑞,𝑝(𝛼, 𝑘) of 𝑘-uniformly 𝑝-valent convex with respect to 𝑞-differentiation of order 𝛼 (−1 ≤ 𝛼 < 𝑝) if it satisfies

R {1 +𝑧𝐷 2 𝑞(𝑓 (𝑧)) 𝐷_𝑞(𝑓 (𝑧)) − 𝛼} ≥ 𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨1 + 𝑧𝐷2 𝑞(𝑓 (𝑧)) 𝐷_𝑞(𝑓 (𝑧)) − [𝑝]𝑞󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨, (𝑘 ≥ 0, 𝑧 ∈ U) . (17) In the following, we define the fractional 𝑞-calculus operators of a complex-valued function 𝑓(𝑧), which were recently studied by Purohit and Raina [5].

Definition 1 (fractionalintegral operator). The fractional

𝑞-integral operator𝐼_𝑞,𝑧𝛿 of a function𝑓(𝑧) of order 𝛿 is defined by 𝐼_𝑞,𝑧𝛿 𝑓 (𝑧) ≡ 𝐷−𝛿_𝑞,𝑧𝑓 (𝑧) =_Γ1 𝑞(𝛿)∫ 𝑧 0 (𝑧 − 𝑡𝑞)𝛿−1𝑓 (𝑡) 𝑑𝑞𝑡, (𝛿 > 0) , (18)

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where𝑓(𝑧) is analytic in a simply connected region of the 𝑧-plane containing the origin and the𝑞-binomial function (𝑧 − 𝑡𝑞)_𝛿−1is given by

(𝑧 − 𝑡𝑞)_𝛿−1= 𝑧𝛿−1₁Φ₀[𝑞−𝛿+1; −; 𝑞,𝑡𝑞𝛿

𝑧 ] . (19) The series ₁Φ₀[𝛿; −; 𝑞, 𝑧] is single valued when | arg(𝑧)| < 𝜋 and |𝑧| < 1 (see for details [2], pp. 104–106); there-fore, the function(𝑧 − 𝑡𝑞)_𝛿−1 in (18) is single valued when | arg(−𝑡𝑞𝛿_{/𝑧)| < 𝜋, |𝑡𝑞}𝛿_{/𝑧| < 1, and | arg(𝑧)| < 𝜋.}

Definition 2 (fractional𝑞-derivative operator). The fractional

𝑞-derivative operator 𝐷𝛿_𝑞,𝑧 of a function𝑓(𝑧) of order 𝛿 is defined by 𝐷𝛿_𝑞,𝑧𝑓 (𝑧) ≡ 𝐷𝑞,𝑧 𝐼𝑞,𝑧1−𝛿𝑓 (𝑧) = _Γ 1 𝑞(1 − 𝛿) × 𝐷_𝑞,𝑧∫𝑧 0 (𝑧 − 𝑡𝑞)−𝛿𝑓 (𝑡) 𝑑𝑞𝑡, (0 ≤ 𝛿 < 1) , (20) where 𝑓(𝑧) is suitably constrained and the multiplicity of (𝑧 − 𝑡𝑞)_−𝛿is removed as inDefinition 1.

Definition 3 (extended fractional 𝑞-derivative operator).

Under the hypotheses of Definition 2, the fractional 𝑞-derivative for a function𝑓(𝑧) of order 𝛿 is defined by

𝐷𝛿_𝑞,𝑧𝑓 (𝑧) = 𝐷_𝑞,𝑧𝑚𝐼_𝑞,𝑧𝑚−𝛿𝑓 (𝑧) , (21) where𝑚 − 1 ≤ 𝛿 < 1, 𝑚 ∈ N₀ = N ∪ {0}, and N denotes the set of natural numbers.

Remark 4. It follows fromDefinition 2that

𝐷𝛿_𝑞,𝑧𝑧𝑛 =_ΓΓ𝑞(𝑛 + 1)

𝑞(𝑛 + 1 − 𝛿)𝑧

𝑛−𝛿 _{(𝛿 ≥ 0, 𝑛 > −1) .} ₍₂₂₎

2. The Operator

D

𝛿,𝑚_{𝑞,𝑝,𝜆}

Using𝐷_𝑞,𝑧𝛿 , we define a𝑞-differintegral operator Ω𝛿_𝑞,𝑝: A_𝑝 → A_𝑝as follows: Ω𝛿 𝑞,𝑝𝑓 (𝑧) = Γ_𝑞(𝑝 + 1 − 𝛿) Γ_𝑞(𝑝 + 1) 𝑧𝛿𝐷𝛿𝑞,𝑧𝑓 (𝑧) = 𝑧𝑝+ ∑∞ 𝑛=𝑝+1 Γ_𝑞(𝑝 + 1 − 𝛿) Γ_𝑞(𝑛 + 1) Γ𝑞(𝑝 + 1) Γ𝑞(𝑛 + 1 − 𝛿) 𝑎𝑛𝑧 𝑛_, (𝛿 < 𝑝 + 1; 𝑛 ∈ N; 0 < 𝑞 < 1; 𝑧 ∈ U) , (23)

where𝐷𝛿_𝑞,𝑧𝑓(𝑧) in (23) represents, respectively, a fractional 𝑞-integral of𝑓(𝑧) of order 𝛿 when −∞ < 𝛿 < 0 and a fractional 𝑞-derivative of 𝑓(𝑧) of order 𝛿 when 0 ≤ 𝛿 < 𝑝 + 1. Here we note thatΩ0_𝑞,𝑝𝑓(𝑧) = 𝑓(𝑧).

We now define a linear multiplier fractional 𝑞-differintegral operatorD𝛿,𝑚_{𝑞,𝑝,𝜆}as follows:

D𝛿,0_{𝑞,𝑝,𝜆}𝑓 (𝑧) = 𝑓 (𝑧) , D𝛿,1_{𝑞,𝑝,𝜆}𝑓 (𝑧) = (1 − 𝜆) Ω𝛿𝑞,𝑝𝑓 (𝑧) + 𝜆 𝑧 [𝑝]_𝑞𝐷𝑞(Ω𝛿𝑞,𝑝𝑓 (𝑧)) , (𝜆 ≥ 0) , D𝛿,2_{𝑞,𝑝,𝜆}𝑓 (𝑧) = D𝛿,1_{𝑞,𝑝,𝜆}(D𝛿,1_{𝑞,𝑝,𝜆}𝑓 (𝑧)) , .. . D𝛿,𝑚_{𝑞,𝑝,𝜆}𝑓 (𝑧) = D𝛿,1𝑞,𝑝,𝜆(D𝛿,𝑚−1𝑞,𝑝,𝜆 𝑓 (𝑧)) , 𝑚 ∈ N. (24)

If𝑓(𝑧) ∈ A_𝑝is given by (1), then by (24) we have

D𝛿,𝑚_{𝑞,𝑝,𝜆}𝑓 (𝑧) = 𝑧𝑝+ ∑∞ 𝑛=𝑝+1 (Γ𝑞(𝑝 + 1 − 𝛿) Γ𝑞(𝑛 + 1) Γ_𝑞(𝑝 + 1) Γ_𝑞(𝑛 + 1 − 𝛿)[1 − 𝜆 + [𝑛]𝑞 [𝑝]_𝑞𝜆]) 𝑚 × 𝑎_𝑛𝑧𝑛. (25)

It can be seen that, by specializing the parameters, the operatorD𝛿,𝑚_{𝑞,𝑝,𝜆}reduces to many known and new integral and differential operators. In particular, when𝛿 = 0, 𝑝 = 1, and 𝑞 → 1 the operator D𝛿,𝑚_{𝑞,𝑝,𝜆}reduces to the operator introduced by AL-Oboudi [13] and if𝛿 = 0, 𝑝 = 1, 𝜆 = 1, and 𝑞 → 1 it reduces to the operator introduced by Sˇalˇagean [14].

By using the operatorD𝛿,𝑚_{𝑞,𝑝,𝜆}𝑓(𝑧) defined by (24) and 𝑞-differentiation, we introduce two new subclasses of analytic functionsUS𝛿,𝑚_{𝑞,𝑝,𝜆}(𝛼, 𝑘) and UC𝛿,𝑚_{𝑞,𝑝,𝜆}(𝛼, 𝑘) as follows.

A function𝑓 ∈ A_𝑝is said to be in the classUS𝛿,𝑚_{𝑞,𝑝,𝜆}(𝛼, 𝑘) if and only if R{_{ { 𝑧𝐷_𝑞(D𝛿,𝑚 𝑞,𝑝,𝜆𝑓 (𝑧)) D𝛿,𝑚 𝑞,𝑝,𝜆𝑓 (𝑧) − 𝛼}_} } ≥ 𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑧𝐷_𝑞(D𝛿,𝑚_{𝑞,𝑝,𝜆}𝑓 (𝑧)) D𝛿,𝑚_{𝑞,𝑝,𝜆}𝑓 (𝑧) − [𝑝]𝑞 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 , (−1 ≤ 𝛼 < 𝑝, 𝑘 ≥ 0) . (26)

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Furthermore, a function𝑓 ∈ A_𝑝is said to be in the class UC𝛿,𝑚 𝑞,𝑝,𝜆(𝛼, 𝑘) if and only if R{_{ { 1 + 𝑧𝐷 2 𝑞(D𝛿,𝑚𝑞,𝑝,𝜆𝑓 (𝑧)) 𝐷_𝑞(D𝛿,𝑚 𝑞,𝑝,𝜆𝑓 (𝑧)) − 𝛼}_} } ≥ 𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 1 +𝑧𝐷 2 𝑞(D𝛿,𝑚𝑞,𝑝,𝜆𝑓 (𝑧)) 𝐷_𝑞(D𝛿,𝑚 𝑞,𝑝,𝜆𝑓 (𝑧)) − [𝑝]_𝑞󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 , (−1 ≤ 𝛼 < 𝑝, 𝑘 ≥ 0) . (27) It is interesting to note that the classes US𝛿,𝑚_{𝑞,𝑝,𝜆}(𝛼, 𝑘) andUC𝛿,𝑚_{𝑞,𝑝,𝜆}(𝛼, 𝑘) generalize several well-known subclasses of analytic functions. For instance, if𝑞 → 1, then

(1)US𝛿,0_1,𝑝,𝜆(𝛼, 𝑘) = US_𝑝(𝛼, 𝑘),

(2)US0,1_1,𝑝,1(𝛼, 𝑘) = UC𝛿,0_1,𝑝,𝜆(𝛼, 𝑘) = UC_𝑝(𝛼, 𝑘), (3)US𝛿,0_1,𝑝,𝜆(𝛼, 0) = S∗_𝑝(𝛼),

(4)US0,1_1,𝑝,1(𝛼, 0) = UC𝛿,0_1,𝑝,𝜆(𝛼, 0) = C_𝑝(𝛼).

3. The

𝑝-Valent 𝑞-Integral Operators 𝐹

_𝑞

and

𝐺

_𝑞

We now introduce two new𝑝-valent 𝑞-integral operators as follows.

Definition 5. Let 𝑚 = (𝑚₁, 𝑚₂, . . . , 𝑚_𝑛) ∈ N𝑛₀, 𝛾 =

(𝛾₁, 𝛾₂, . . . , 𝛾_𝑛) ∈ R𝑛₊ and 𝑓_𝑖 ∈ A_𝑝 for all𝑖 = {1, 2, . . . , 𝑛}, 𝑛 ∈ N. Then 𝐹_𝑞(𝑧) : A𝑛_𝑝 → A_𝑝is defined as 𝐹_𝑞(𝑧) = F𝛿,𝛾,𝑚_{𝑞,𝑝,𝜆} (𝑓₁, 𝑓₂, . . . , 𝑓_𝑛) = ∫𝑧 0 [𝑝]𝑞 𝑡 𝑝−1_∏𝑛 𝑖=1 (D 𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑡) 𝑡𝑝 ) 𝛾𝑖 𝑑_𝑞𝑡, (28) and𝐺_𝑞(𝑧) : A𝑛_𝑝 → A_𝑝is defined as 𝐺_𝑞(𝑧) = G𝛿,𝛾,𝑚_{𝑞,𝑝,𝜆}(𝑓₁, 𝑓₂, . . . , 𝑓_𝑛) = ∫𝑧 0 [𝑝]𝑞𝑡 𝑝−1_∏𝑛 𝑖=1 (𝐷𝑞(D 𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑡)) [𝑝]_𝑞 𝑡𝑝−1 ) 𝛾𝑖 𝑑_𝑞𝑡, (29) whereD𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑡) is given by (24).

It is interesting to observe that several well-known and new integral operators are special cases of the operators𝐹_𝑞(𝑧) and𝐺_𝑞(𝑧). We list a few of them in the following remarks.

Remark 6. Letting𝑚_𝑖= 0 for all 𝑖 = {1, 2, . . . , 𝑛} and 𝑞 → 1,

the𝑞-integral operator 𝐹_𝑞(𝑧) reduces to the operator 𝐹_𝑝(𝑧) studied by Frasin in [15]. Upon setting𝑝 = 1, 𝛿 = 0, 𝜆 = 1, and𝑞 → 1, we obtain the integral operator 𝐷𝑘𝐹(𝑧) studied by Breaz et al. in [16]. For𝑝 = 1, 𝑚₁ = 𝑚₂ = ⋅ ⋅ ⋅ = 𝑚_𝑛 = 0,

and𝑞 → 1, the operator 𝐹_𝑞(𝑧) reduces to the operator 𝐹_𝑛(𝑧) which was studied by D. Breaz and N. Breaz in [17]. Observe that when𝑝 = 𝑛 = 1, 𝑚₁ = 0, 𝛾₁= 𝛾, and 𝑞 → 1, we obtain the integral operator𝐼_𝛾(𝑓)(𝑧) studied by Pescar and Owa in [18]. Also, for𝑝 = 𝑛 = 1, 𝑚₁ = 0, 𝛾₁ = 1, and 𝑞 → 1, the 𝑞-integral operator 𝐹𝑞(𝑧) reduces to the Alexander integral

operator𝐼(𝑓)(𝑧) studied in [19].

Remark 7. Letting𝑚_𝑖= 0 for all 𝑖 = {1, 2, . . . , 𝑛} and 𝑞 → 1,

the𝑞-integral operator 𝐺_𝑞(𝑧) reduces to the operator 𝐺_𝑝(𝑧) studied by Frasin in [15]. For𝑝 = 1, 𝑚₁= 𝑚₂= ⋅ ⋅ ⋅ = 𝑚_𝑛 = 0 and 𝑞 → 1, the operator 𝐺_𝑞(𝑧) reduces to the operator 𝐺_𝛾₁_,𝛾₂_,...,𝛾_𝑛(𝑧) which was studied by Breaz et al. (see [20]). Also, for𝑝 = 𝑛 = 1, 𝑚₁= 0, 𝛾₁= 1, and 𝑞 → 1, the 𝑞-integral oper-ator𝐺_𝑞(𝑧) reduces to the integral operator 𝐺(𝑧) introduced and studied by Pfaltzgra (see [21]).

In this paper, we obtain the order of convexity with respect to𝑞-differentiation of the 𝑞-integral operators 𝐹_𝑞(𝑧) and 𝐺_𝑞(𝑧) on the classes US𝛿,𝑚_{𝑞,𝑝,𝜆}(𝛼, 𝑘) and UC𝛿,𝑚_{𝑞,𝑝,𝜆}(𝛼, 𝑘). As special cases, the order of convexity of the operators ∫₀𝑧(𝑓(𝑡)/𝑡)𝛾𝑑𝑡 and ∫₀𝑧(𝑓󸀠(𝑡))𝛾𝑑𝑡 is also given.

4. Convexity of the Operator

𝐹

_𝑞

First, we prove the following convexity result with respect to 𝑞-differentiation of the operator 𝐹_𝑞.

Theorem 8. Let 𝑚 = (𝑚₁, 𝑚₂, . . . , 𝑚_𝑛) ∈ N𝑛 0, 𝛾 = (𝛾₁, 𝛾₂, . . . , 𝛾_𝑛) ∈ R𝑛 +,−1 ≤ 𝛼𝑖 < 𝑝, 𝑘𝑖 > 0, and 𝑓𝑖 ∈ US𝛿,𝑚𝑖 𝑞,𝑝,𝜆(𝛼𝑖, 𝑘𝑖) for all 𝑖 = {1, 2, . . . , 𝑛}, 𝑛 ∈ N. If 0 ≤ 𝑝 +∑𝑛 𝑖=1 𝛾_𝑖(𝛼_𝑖− 𝑝) < 𝑝, (30)

then the𝑞-integral operator 𝐹_𝑞(𝑧) defined by (28) is𝑝-valently

convex with respect to𝑞-differentiation of order 𝑝+∑𝑛_𝑖=1𝛾_𝑖(𝛼_𝑖−

𝑝).

Proof. From (28), we observe that𝐹_𝑞(𝑧) ∈ A_𝑝. On the other

hand, it is easy to verify that

𝐷𝑞(𝐹𝑞(𝑧)) = [𝑝]𝑞𝑧𝑝−1 𝑛 ∏ 𝑖=1( D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧) 𝑧𝑝 ) 𝛾𝑖 . (31) Now by logarithmic𝑞-differentiation we have

ln𝑞 𝑞 − 1 𝐷2 𝑞(𝐹𝑞(𝑧)) 𝐷𝑞(𝐹𝑞(𝑧)) = ln𝑞 𝑞 − 1[ [ 𝑝 − 1 𝑧 + 𝑛 ∑ 𝑖=1 𝛾_𝑖(𝐷𝑞(D 𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧) −𝑝 𝑧)] ] . (32)

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Therefore, 1 +𝑧𝐷 2 𝑞(𝐹𝑞(𝑧)) 𝐷_𝑞(𝐹_𝑞(𝑧)) = 𝑝 + 𝑛 ∑ 𝑖=1 𝛾_𝑖(𝑧𝐷𝑞(D 𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧) − 𝛼_𝑖) +∑𝑛 𝑖=1 𝛾_𝑖(𝛼_𝑖− 𝑝) . (33) Taking the real parts on both sides of the above equation, we have R{_{ { 1 +𝑧𝐷 2 𝑞(𝐹𝑞(𝑧)) 𝐷_𝑞(𝐹_𝑞(𝑧)) } } } = 𝑝 +∑𝑛 𝑖=1 𝛾_𝑖(𝛼_𝑖− 𝑝) +∑𝑛 𝑖=1 𝛾_𝑖R (𝑧𝐷𝑞(D 𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧) − 𝛼_𝑖) . (34) Since𝑓_𝑖∈ US𝛿,𝑚𝑖

𝑞,𝑝,𝜆(𝛼𝑖, 𝑘𝑖) for all 𝑖 = {1, 2, . . . , 𝑛}, from (26) we

get R{_{ { 1 + 𝑧𝐷 2 𝑞(𝐹𝑞(𝑧)) 𝐷_𝑞(𝐹_𝑞(𝑧)) } } } ≥ 𝑝 +∑𝑛 𝑖=1 𝛾_𝑖(𝛼_𝑖− 𝑝) +∑𝑛 𝑖=1 𝛾_𝑖 𝑘_𝑖󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑧𝐷_𝑞(D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧) − [𝑝]_𝑞󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 . (35) As∑𝑛_𝑖=1𝛾_𝑖 𝑘_𝑖|(𝑧𝐷_𝑞(D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧))/D𝛿,𝑚𝑞,𝑝,𝜆𝑖𝑓𝑖(𝑧)) − [𝑝]𝑞| > 0, for

all𝑖 = {1, 2, . . . , 𝑛}, we obtain from the above R{_{ { 1 +𝑧𝐷 2 𝑞(𝐹𝑞(𝑧)) 𝐷𝑞(𝐹𝑞(𝑧)) } } } > 𝑝 +∑𝑛 𝑖=1𝛾𝑖(𝛼𝑖− 𝑝) . (36)

This completes the proof.

Corollary 9. Let 𝑚 = (𝑚₁, 𝑚₂, . . . , 𝑚_𝑛) ∈ N𝑛 0, 𝛾 = (𝛾₁, 𝛾₂, . . . , 𝛾_𝑛) ∈ R𝑛 +,−1 ≤ 𝛼𝑖 < 𝑝, 𝑘𝑖 > 0 and 𝑓𝑖 ∈ US𝛿,𝑚𝑖 𝑞,𝑝,𝜆(𝛼𝑖, 𝑘𝑖) for all 𝑖 = {1, 2, . . . , 𝑛}, 𝑛 ∈ N. If 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 𝑧𝐷_𝑞(D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧) − [𝑝]_𝑞󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 > −𝑝 + ∑𝑛𝑖=1𝛾𝑖(𝛼𝑖− 𝑝) ∑𝑛_𝑖=1𝛾_𝑖𝑘_𝑖 , (37)

for all 𝑖 = {1, 2, . . . , 𝑛}, then the 𝑞-integral operator 𝐹_𝑞(𝑧)

defined by (28) is 𝑝-valently convex with respect to

𝑞-differentiation inU.

Proof. From (35) and (37) we easily find that𝐹_𝑞 ∈ C_𝑞,𝑝.

Letting𝑞 → 1, 𝑝 = 𝑛 = 1, 𝑚₁= 0, 𝛾₁= 𝛾, 𝛼₁= 𝛼, 𝑘₁= 𝑘, and𝑓₁= 𝑓 inTheorem 8, we have the following.

Corollary 10. Let 𝛾 > 0, −1 ≤ 𝛼 < 1, 𝑘 > 0, and 𝑓 ∈

UST(𝛼, 𝑘). If 0 ≤ 1 + 𝛾(𝛼 − 1) < 1, then the integral operator ∫₀𝑧(𝑓(𝑡)/𝑡)𝛾𝑑𝑡 is convex of order 1 + 𝛾(𝛼 − 1) in U.

5. Convexity of the Operator

𝐺

_𝑞

Now, we prove the following convexity result with respect to 𝑞-differentiation of the operator 𝐺_𝑞.

Theorem 11. Let 𝑚 = (𝑚₁, 𝑚₂, . . . , 𝑚_𝑛) ∈ N𝑛 0, 𝛾 = (𝛾₁, 𝛾₂, . . . , 𝛾_𝑛) ∈ R𝑛 +,−1 ≤ 𝛼𝑖 < 𝑝, 𝑘𝑖 > 0, and 𝑓𝑖 ∈ UC𝛿,𝑚𝑖 𝑞,𝑝,𝜆(𝛼𝑖, 𝑘𝑖) for all 𝑖 = {1, 2, . . . , 𝑛}, 𝑛 ∈ N. If 0 ≤ 𝑝 +∑𝑛 𝑖=1 𝛾𝑖(𝛼𝑖− 𝑝) < 𝑝, (38)

then the𝑞-integral operator 𝐺_𝑞(𝑧) defined by (29) is𝑝-valently

convex with respect to𝑞-differentiation of order 𝑝+∑𝑛_𝑖=1𝛾_𝑖(𝛼_𝑖−

𝑝).

Proof. From (29), we observe that𝐺_𝑞(𝑧) ∈ A_𝑝. On the other

hand, it is easy to verify that

𝐷_𝑞(𝐺_𝑞(𝑧)) = [𝑝]_𝑞 𝑧𝑝−1∏𝑛 𝑖=1 (𝐷𝑞(D 𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) [𝑝]_𝑞𝑧𝑝−1 ) 𝛾𝑖 . (39) Now by logarithmic𝑞-differentiation we have

ln𝑞 𝑞 − 1 𝐷2 𝑞(𝐺𝑞(𝑧)) 𝐷_𝑞(𝐺_𝑞(𝑧)) = ln𝑞 𝑞 − 1[ [ 𝑝 − 1 𝑧 + 𝑛 ∑ 𝑖=1 𝛾_𝑖(𝐷 2 𝑞(D𝛿,𝑚𝑞,𝑝,𝜆𝑖𝑓𝑖(𝑧)) 𝐷𝑞(D𝛿,𝑚𝑞,𝑝,𝜆𝑖𝑓𝑖(𝑧)) −𝑝 − 1 𝑧 )] ] . (40) Therefore, 1 + 𝑧𝐷 2 𝑞(𝐺𝑞(𝑧)) 𝐷_𝑞(𝐺_𝑞(𝑧)) = 𝑝 +∑𝑛 𝑖=1 𝛾_𝑖(1 +𝐷 2 𝑞(D𝛿,𝑚𝑞,𝑝,𝜆𝑖𝑓𝑖(𝑧)) 𝐷_𝑞(D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) − 𝛼_𝑖) +∑𝑛 𝑖=1 𝛾_𝑖(𝛼_𝑖− 𝑝) . (41)

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Taking the real parts on both sides of the above equation, we have R{_{ { 1 +𝑧𝐷 2 𝑞(𝐺𝑞(𝑧)) 𝐷_𝑞(𝐺_𝑞(𝑧)) } } } = 𝑝 +∑𝑛 𝑖=1 𝛾_𝑖(𝛼_𝑖− 𝑝) +∑𝑛 𝑖=1 𝛾_𝑖R (1 +𝐷 2 𝑞(D𝛿,𝑚𝑞,𝑝,𝜆𝑖𝑓𝑖(𝑧)) 𝐷_𝑞(D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) − 𝛼_𝑖) . (42) Since𝑓_𝑖∈ UC𝛿,𝑚𝑖

𝑞,𝑝,𝜆(𝛼𝑖, 𝑘𝑖) for all 𝑖 = {1, 2, . . . , 𝑛}, from (27) we

get R{_{ { 1 +𝑧𝐷 2 𝑞(𝐺𝑞(𝑧)) 𝐷_𝑞(𝐺_𝑞(𝑧)) } } } ≥ 𝑝 +∑𝑛 𝑖=1 𝛾_𝑖(𝛼_𝑖− 𝑝) +∑𝑛 𝑖=1 𝛾_𝑖𝑘_𝑖󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 1 + 𝐷 2 𝑞(D𝛿,𝑚𝑞,𝑝,𝜆𝑖𝑓𝑖(𝑧)) 𝐷_𝑞(D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) − [𝑝]_𝑞󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 . (43) As∑𝑛_𝑖=1𝛾_𝑖 𝑘_𝑖|1 + (𝐷2_𝑞(D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧))/𝐷𝑞(D𝛿,𝑚𝑞,𝑝,𝜆𝑖𝑓𝑖(𝑧))) − [𝑝]𝑞| >

0, for all 𝑖 = {1, 2, . . . , 𝑛}, we obtain from the above R{_{ { 1 +𝑧𝐷 2 𝑞(𝐺𝑞(𝑧)) 𝐷_𝑞(𝐺_𝑞(𝑧)) } } } > 𝑝 +∑𝑛 𝑖=1 𝛾_𝑖(𝛼_𝑖− 𝑝) . (44) This completes the proof.

Corollary 12. Let 𝑚 = (𝑚₁, 𝑚₂, . . . , 𝑚_𝑛) ∈ N𝑛 0, 𝛾 = (𝛾₁, 𝛾₂, . . . , 𝛾_𝑛) ∈ R𝑛 +,−1 ≤ 𝛼𝑖 < 𝑝, 𝑘𝑖 > 0, and 𝑓𝑖 ∈ UC𝛿,𝑚𝑖 𝑞,𝑝,𝜆(𝛼𝑖, 𝑘𝑖) for all 𝑖 = {1, 2, . . . , 𝑛}, 𝑛 ∈ N. If 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 1 +𝐷 2 𝑞(D𝛿,𝑚𝑞,𝑝,𝜆𝑖𝑓𝑖(𝑧)) 𝐷_𝑞(D𝛿,𝑚𝑖 𝑞,𝑝,𝜆𝑓𝑖(𝑧)) − [𝑝]_𝑞󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 > −𝑝 + ∑𝑛𝑖=1𝛾𝑖(𝛼𝑖− 𝑝) ∑𝑛_𝑖=1𝛾_𝑖 𝑘_𝑖 , (45)

for all 𝑖 = {1, 2, . . . , 𝑛}, then the 𝑞-integral operator 𝐺_𝑞(𝑧)

defined by (29) is 𝑝-valently convex with respect to

𝑞-differentiation inU.

Proof. From (43) and (45) we easily find that𝐺_𝑞∈ C_𝑞,𝑝.

Letting𝑞 → 1, 𝑝 = 𝑛 = 1, 𝑚₁= 0, 𝛾₁= 𝛾, 𝛼₁= 𝛼, 𝑘₁= 𝑘, and𝑓₁= 𝑓 inTheorem 11, we have the following.

Corollary 13. Let 𝛾 > 0, −1 ≤ 𝛼 < 1, 𝑘 > 0, and 𝑓 ∈

UCV(𝛼, 𝑘). If 0 ≤ 1 + 𝛾(𝛼 − 1) < 1, then the integral operator ∫₀𝑧(𝑓󸀠_(𝑡))𝛾_{𝑑𝑡 is convex of order 1 + 𝛾(𝛼 − 1) in U.}

We remark in conclusion that, by suitably specializing the parameters in Theorems8and11, we can deduce the results obtained in [15,22,23].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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