## A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus Operator

### BibTeX

@MISC{Khairnar_asubclass,

author = {S. M. Khairnar and Meena More},

title = {A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus Operator},

year = {}

}

### OpenURL

### Abstract

AbstractIn this paper, we introduce a new class K µ,γ,η (α, β) of uniformly convex functions defined by a certain fractional calculus operator. The class has interesting subclasses like β-uniformly starlike, β-uniformly convex and β-uniformly pre-starlike functions. Properties like coefficient estimates, growth and distortion theorems, modified Hadamard product, inclusion property, extreme points, closure theorem and other properties of this class are studied. Lastly, we discuss a class preserving integral operator, radius of starlikeness, convexity and close-toconvexity and integral mean inequality for functions in the class K µ,γ,η (α, β).

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Citation Context ... is the Pochhammer symbol defined by Γ(a + k) (a)k = Γ(a) = { 1 : k = 0 a(a + 1)(a + 2) · · · (a + k − 1) : k ∈ IN We note that L(a,c)f(z) = φ(a,b;z) ∗ f(z), for f ∈ S is the Carlson-Shaffer operator =-=[1]-=-, which is a special case of the Dziok-Srivastava operator [2]. Following Saigo [15] the fractional integral and derivative operators involving the Gauss’s hypergeometric function 2F1(a,b;c;z) are def... |

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Citation Context ...,0) = K(α), where S(α) and K(α) are respectively the popular classes of starlike and convex functions of order α (0 ≤ α < 1). The classes UST(α,β) and UCV (α,β) were introduced and studied by Goodman =-=[4]-=-, Rønning [13] and Minda and Ma [8]. Clearly f ∈ UCV (α,β) if and only if zf ′ ∈ UST(α,β).A function f(z) is said to be close-to-convex of order r, 0 ≤ r < 1 if Ref ′ (z) > r. Let φ(a,c;z) be the inco... |

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Citation Context ...here S(α) and K(α) are respectively the popular classes of starlike and convex functions of order α (0 ≤ α < 1). The classes UST(α,β) and UCV (α,β) were introduced and studied by Goodman [4], Rønning =-=[13]-=- and Minda and Ma [8]. Clearly f ∈ UCV (α,β) if and only if zf ′ ∈ UST(α,β).A function f(z) is said to be close-to-convex of order r, 0 ≤ r < 1 if Ref ′ (z) > r. Let φ(a,c;z) be the incomplete beta fu... |

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Citation Context ... instance, λ(t) = (1 + c)tc ,c > −1, for which Lλ is known as the Bernardi operator. For λ(t) = 2δ 1 t(log Γ(δ) t )δ−1, δ ≥ 0 (5.1) we get the integral operator introduced by Jung, Kim and Srivastava =-=[6]-=-. Let us consider the function λ(t) = (c + 1)δ t Γ(δ) c (log 1 t )δ−1, c > −1, δ ≥ 0. (5.2) Notice that for c = 1 we get the integral operator introduced by Jung, Kim and Srivastava. We next show that... |

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Citation Context ...η 0,z f(z)) ′ | − 1| ≤ (2.4) c(1 − α)(2 − γ)(2 − µ + η) a(β − α + 2)(2 − γ + η) |z| (2.5) Note that for a = c = 1;β = 1, we get the result obtained by G. Murugusundaramoorthy, T. Rosy and M. Darus in =-=[9]-=-. The bounds in (2.4) and (2.5), are attained for the function f(z) = z − c(1 − α)(2 − γ)(2 − µ + η) 2a(β − α + 2)(2 − γ + η) z2 3 Characterization Property Theorem 3.1. Let µ,γ,η ∈ IR such that µ(−∞ ... |

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Citation Context ...oved by requiring log(z − t) to be real when (z − t) > 0 and is well defined in the unit disc. Notice that J µ,µ,η 0,z f(z) = D µ 0,zf(z) which is the well known fractional derivative operator by Owa =-=[10]-=-. The fractional operator U µ,γ,η 0,z is defined in terms of J µ,γ,η 0,z for convenience as follows U µ,γ,η 0,z f(z) = Γ(2 − γ)Γ(2 − µ + η) z Γ(2 − γ + η) γ J µ,γ,η 0,z f(z) (1.8) (−∞ < µ < 1; −∞ < γ ... |

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