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A Subclass of Uniformly Convex Functions
Associated with Certain Fractional Calculus
Operator
S. M. Khairnar and Meena More∗
AbstractIn this paper, we introduce a new class
Kµ,γ,η(α,β) of uniformly convex functions defined by
a certain fractional calculus operator.
has interesting subclasses like βuniformly starlike, β
uniformly convex and βuniformly prestarlike func
tions.Properties like coefficient estimates, growth
and distortion theorems, modified Hadamard prod
uct, inclusion property, extreme points, closure the
orem and other properties of this class are studied.
Lastly, we discuss a class preserving integral oper
ator, radius of starlikeness, convexity and closeto
convexity and integral mean inequality for functions
in the class Kµ,γ,η(α,β).
The class
Keywords and Phrases: Fractional derivative, Univa
lent function, Uniformly convex function, Fractional
integral operator, Incomplete beta function, Modified
Hadamard product.
1 Introduction
Let S denote the class of functions of the form
f(z) = z +
∞
?
k=2
akzk
(1.1)
which are analytic and univalent in the unit disc U = {z :
z < 1}. Also denote by T the class of functions of the
form
f(z) = z −
∞
?
k=2
akzk
(ak≥ 0,z ∈ U)(1.2)
which are analytic and univalent in U.
For g(z) = z −
∞
?
k=2
bkzkthe modified Hadamard product
of f(z) and g(z) is defined by
(f ∗ g)(z) = z −
∞
?
k=2
akbkzk.(1.3)
∗Department
Engineering,
smkhairnar2007@gmail.com, meenamores@gmail.com
of
Alandi
Mathematics,

Maharashtra
Pune
Academy
S.),of412105,(M.India,
A function f(z) ∈ S is said to be βuniformly starlike of
order α,(−1 ≤ α < 1),β ≥ 0 and (z ∈ U), denoted by
UST(α,β), if and only if
Re
?zf′(z)
f(z)
− α
?
≥ β
????
zf′(z)
f(z)
− 1
????
(1.4)
A function f(z) ∈ S is said to be βuniformly convex of
order α,(−1 ≤ α < 1),β ≥ 0 and (z ∈ U), denoted by
UCV (α,β), if and only if
Re
?
1 +zf′′(z)
f′(z)
− α
?
≥ β
????
zf′′(z)
f′(z)
????
(1.5)
Notice that, UST(α,0) = S(α) and UCV (α,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 closetoconvex of order r, 0 ≤
r < 1 if Ref′(z) > r. Let φ(a,c;z) be the incomplete
beta function defined by
φ(a,c;z)=z +
∞
?
k=2
(a)k−1
(c)k−1zk(a ?= −1,−2,−3,···
and c ?= 0,−1,−2,−3,···) (1.6)
where (a)kis the Pochhammer symbol defined by
(a)k=Γ(a + k)
Γ(a)
=
?
1
a(a + 1)(a + 2)···(a + k − 1)
:
:
k = 0
k ∈ I N
We note that L(a,c)f(z) = φ(a,b;z) ∗ f(z), for f ∈ S is
the CarlsonShaffer operator [1], which is a special case
of the DziokSrivastava operator [2].
Following Saigo [15] the fractional integral and derivative
operators involving the Gauss’s hypergeometric function
2F1(a,b;c;z) are defined as follows.
IAENG International Journal of Applied Mathematics, 39:3, IJAM_39_07
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(Advanced on line publication:1 August 2009)
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Setting
1 +
∞
?
k=2
h(k)akzk−1= 1 +
(1 − α)
[j(1 + β) − (α + β)]w(z)j−1.
We note that
(w(z))j−1=[j(1 + β) − (α + β)]
(1 − α)
∞
?
k=2
h(k)akzk−1, (9.7)
and w(0) = 0.
function w(z) satisfies w(z) < 1,z ∈ U
Moreover, we prove that the analytic
w(z)j−1
≤
?????
[j(1 + β) − (α + β)]
(1 − α)
∞
?
∞
?
k=2
h(k)akzk−1
?????
≤
[j(1 + β) − (α + β)]
(1 − α)
k=2
h(k)akzk−1
≤z[j(1 + β) − (α + β)]
(1 − α)
h(2)
∞
?
k=2
ak
≤z < 1 by hypothesis (9.1).
This completes the proof of Theorem 9.2.
As a particular case of Theorem 9.2 we can derive the
result for the function f(z) by taking a = c = 1 and
µ = γ = 0 and thus Mµ,γ,η
0,z
f(z) = f(z) .
References
[1] Carlson, B. C. and Shaffer, D. B. Starlike and pre
starlike hypergeometric functions, SIAM J. Math.
Anal., 15 (1984), pp. 737745.
[2] Dziok, J. and Srivastava, H. M., Classes of an
lytic functions associated with the generalized hy
pergeometric function, Appl. Math. Comput., 103(1)
(1999), pp. 113.
[3] Ghanim, F. and Darus, M., On new subclass of an
alytic pvalent function with negative coefficient for
operator on Hilbert space, International Mathemat
ical Forum, 3 (2008) No. 2, pp. 6977.
[4] Goodman, A. W., On uniformly convex functions,
Ann. Polon. Math., 56 (1991), pp. 8792.
[5] Hohlov, J. E., Operators and operations on the class
of univalent functions, Izv. Vyssh. Uchebn. Zaved.
Mat., 197(10) (1978), pp. 8389.
[6] Jung, I. B., Kim, Y. C. and Srivastava, H. M., The
hardly space of analytic functions associated with
certain oneparameter families of integral transform,
J. Math. Anal. Appl., 179 (1993), pp. 138147.
[7] Kim, Y. C. and Rønning, F., Integral transform of
certain subclasses of analytic functions, J. Math.
Anal. Appl., 258 (2001), pp. 466489.
[8] Minda, D. and MA, W., Uniformly convex functions,
Ann. Polon. Math., 57 (1992), pp. 165175.
[9] Murugusundaramoorthy, G., Rosy, T. and Darus,
M., A subclass of uniformly convex functions asso
ciated with certain fractional calculus operators, J.
Inequal. Pure and Appl. Math., 6 (2005), No. 3, Ar
ticle 86, [online : http: //jipam.vu.edu.au].
[10] Owa, S., On the distortion theorems.I, Kyungpook
Mathematical Journal, 18(1) (1978), pp. 5359.
[11] Owa, S. and Uralegaddi, B. A., An application of
the fractional calculus, J. Karnatak Univ. Sci., 32
(1987), pp. 194201
[12] Raina, R. K. and Saigo, M., A note on fractional
calculus operators involving Fox’Hfunction on space
Fp,µ, Recent Advances in Fractional Calculus, Global
Publishing Co. Sauk Rapids, (1993), pp. 219229.
[13] Rønning, F., Uniformly convex functions and a cor
responding class of starlike functions, Proc. Amer.
Math. Soc., 118 (1993), pp. 189196.
[14] Ruscheweyh, S., New Criteria for univalent func
tions,Proc. Amer. Math. Soc., 49 1975, pp. 109115.
[15] Saigo, M., A remark on integral operators involv
ing the Gauss hypergeometric function, Math. Rep.
College General Ed. Kyushu Univ. II, (1978), pp.
135143.
[16] Shenan, J. M., On a subclass of βuniformly con
vex functions defined by DziokSrivastava linear op
erator, Journal of Fundamental Sciences, 3 (2007),
177191.
[17] Shenan, J. M. and Mousa S. Marouf, A subclass
of uniformly convex functions with negative coeffi
cients defined by a certain linear operator, Advances
in Theoretical and Applied Mathematics, 1 (2006),
No. 3, pp. 211222.
[18] Srivastava, H. M., Saigo, M. and Owa, S., A class
of distortion theorems involving certain operators of
fractional calculus, J. Math. Anal., 131 (1988), 412
420.
IAENG International Journal of Applied Mathematics, 39:3, IJAM_39_07
_____________________________________________________________________
(Advanced on line publication:1 August 2009)