Open Math. 2015; 13: 768–780
Open Mathematics
Open Access
Research Article Rabha W. Ibrahim*, Muhammad Zaini Ahmad, and Hiba F. Al-Janaby
Upper and lower bounds of integral operator defined by the fractional hypergeometric function DOI 10.1515/math-2015-0071 Received July 29, 2015; accepted September 10, 2015.
Abstract: In this article, we impose some studies with applications for generalized integral operators for normalized
holomorphic functions. By using the further extension of the extended Gauss hypergeometric functions, new subclasses of analytic functions containing extended Noor integral operator are introduced. Some characteristics of these functions are imposed, involving coefficient bounds and distortion theorems. Further, sufficient conditions for subordination and superordination are illustrated. Keywords: Analytic function, Univalent function, Fractional integral operator, Subordination, Superordination, fractional hypergeometric function, Unit disk MSC: 30C45
1 Introduction The Fractional Calculus treats with derivatives and integrals to an arbitrary order (real and complex order). The calculated characterization of the fractional derivative and integral has been the subject of several areas, such as statistics [1, 2], quantum mechanics [3], economy [4], general physics [5, 6], modeling [7], chaos, fractal and discrete studies [8–14]. There are different studies in complex domain, such as complex fractional differential equations [15–17], fractional integral and differential classes of analytic functions [18–20]. The theory of hypergeometric functions plays an important role in the study of the fractional calculus and the geometric function theory, which underlie the theory of univalent functions. It has gained increasing interest as an active area, and relevant to current research, after its utilization by Branges [21] in the proof of great famous problem in geometric function theory, which is called the Bieberbach’s conjecture. This theory has been developed and enriched with many applications and generalization by prominent complex analysts. Very recently, a great interaction between this attractive area and geometric function theory has motivated renowned authors to actively investigate this field and further extensions of some extended hypergeometric functions have been introduced. For real or complex numbers ˛; ˇ; other than 0; 1; 2; ::: the Gaussian hypergeometric function is defined by (see [21])
*Corresponding Author: Rabha W. Ibrahim: Faculty of Computer Science and Information Technology, University Malaya, 50603 Kuala Lumpur, Malaysia, E-mail:
[email protected] Muhammad Zaini Ahmad: Institute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Malaysia, E-mail:
[email protected] Hiba F. Al-Janaby: Institute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Malaysia, E-mail:
[email protected] © 2015 Ibrahim et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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Upper and lower bounds of integral operator
F .˛; ˇI ; z/ D
1 X ˛ˇ ˛.˛ C 1/ˇ.ˇ C 1/ z 2 .˛/n .ˇ/n n z D1C zC C :::::: . /n .1/n
. C 1/ 2Š
769
(1)
nD0
where .˛/n is the pochhammer symbol defined by ( .˛ C n/ 1; .˛/n D D ˛.˛ C 1/:::.˛ C n .˛/
1/;
nD0 n D f1; 2; :::g
and .i / n.}/n D } Œ.} C 1/n
.}/n
.i i / .}/nC1 D }.} C 1/n D .} C n/.}/n : .a;bI;/ .x; y/; which is defined by Recently, Srivastava et al. [22] have introduced the extended Beta function Bp
a;bI; Bp .x; y/ D
Z1
x
1
.1
/y
1
F
aI bI
p .1
/
d;
(2)
0
0; 0;
