Journal of Inequalities in Pure and Applied Mathematics A SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS ASSOCIATED WITH CERTAIN FRACTIONAL CALCULUS OPERATORS volume 6, issue 3, article 86, 2005.
G. MURUGUSUNDARAMOORTHY, THOMAS ROSY AND MASLINA DARUS Department of Mathematics Vellore Institute of Technology, Deemed University Vellore - 632014, India. EMail:
[email protected] Department of Mathematics Madras Christian College, Chennai - 600059, India. School of Mathematical Sciences Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor, Malaysia. EMail:
[email protected]
Received 03 March, 2005; accepted 26 July, 2005. Communicated by: G. Kohr
Abstract Contents
JJ J
II I
Home Page Go Back Close c
2000 Victoria University ISSN (electronic): 1443-5756 062-05
Quit
Abstract In this paper, we introduce a new class of functions which are analytic and univalent with negative coefficients defined by using a certain fractional calculus and fractional calculus integral operators. Characterization property,the results on modified Hadamard product and integrals transforms are discussed. Further, distortion theorem and radii of starlikeness and convexity are also determined here. 2000 Mathematics Subject Classification: 30C45. Key words: Univalent, Convex, Starlike, Uniformly convex , Fractional calculus integral operator.
Contents 1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Characterization Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Results Involving Modified Hadamard Products . . . . . . . . . . . 12 4 Integral Transform of the Class T R(µ, γ, η, α) . . . . . . . . . . . . 17 References
A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus Operators G. Murugusundaramoorthy, Thomas Rosy and Maslina Darus
Title Page Contents
JJ J
II I
Go Back Close Quit Page 2 of 21 J. Ineq. Pure and Appl. Math. 6(3) Art. 86, 2005
1.
Introduction and Preliminaries
Fractional calculus operators have recently found interesting applications in the theory of analytic functions. The classical definition of fractional calculus and its other generalizations have fruitfully been applied in obtaining, the characterization properties, coefficient estimates and distortion inequalities for various subclasses of analytic functions. Denote by A the class of functions of the form (1.1)
f (z) = z +
∞ X
an z
n
n=2
which are analytic and univalent in the open disc E = {z : z ∈ C and |z| < 1}. Also denote by T [11] the subclass of A consisting of functions of the form (1.2)
f (z) = z −
∞ X
an z n ,
(an ≥ 0).
n=2
A function f ∈ A is said to be in the class of uniformly convex functions of order α, denoted by U CV (α) [9] if 00 zf (z) zf 00 (z) (1.3) Re 1 + 0 − α ≥ β 0 − 1 , f (z) f (z) and is said to be in a corresponding subclass of U CV (α) denote by Sp (α) if 0 0 zf (z) zf (z) − α ≥ β − 1 , (1.4) Re f (z) f (z)
A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus Operators G. Murugusundaramoorthy, Thomas Rosy and Maslina Darus
Title Page Contents
JJ J
II I
Go Back Close Quit Page 3 of 21 J. Ineq. Pure and Appl. Math. 6(3) Art. 86, 2005
where −1 ≤ α ≤ 1 and z ∈ E. The class of uniformly convex and uniformly starlike functions has been extensively studied by Goodman [3, 4] and Ma and Minda [6]. P n If f of the form (1.1) and g(z) = z + ∞ n=2 bn z are two functions in A, then the Hadamard product (or convolution) of f and g is denoted by f ∗ g and is given by (f ∗ g)(z) = z +
(1.5)
∞ X
an b n z n .
n=2
Let φ(a, c; z) be the incomplete beta function defined by (1.6)
φ(a, c; z) = z +
∞ X (a)n n=2
(c)n
n
z ,
c 6= 0, −1, −2, . . . ,
where (λ)n is the Pochhammer symbol defined in terms of the Gamma functions, by ( ) 1 n=0 Γ(λ + n) (λ)n = = Γ(λ) λ(λ + 1)(λ + 2) · · · (λ + n − 1), n ∈ N }
A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus Operators G. Murugusundaramoorthy, Thomas Rosy and Maslina Darus
Title Page Contents
JJ J
II I
Go Back
Further suppose
Close
L(a, c)f (z) = φ(a, c; z) ∗ f (z),
for
where L(a, c) is called Carlson - Shaffer operator [2].
f ∈A
Quit Page 4 of 21 J. Ineq. Pure and Appl. Math. 6(3) Art. 86, 2005
For real number µ (−∞ < µ < 1) and γ (−∞ < γ < 1) and a positive real number η, we define the operator µ,γ,η U0,z : A −→ A
by µ,γ,η U0,z
(1.7)
=z+
∞ X n=2
(2 − γ + η)n−1 (2)n−1 an z n , (2 − γ)n−1 (2 − µ + η)n−1
which for f (z) 6= 0 may be written as Γ(2−γ)Γ(2−µ+γ) µ,γ,η z γ J0,z f (z); 0 ≤ µ < 1 Γ(2−γ+η) µ,γ,η (1.8) U0,z f (z) = Γ(2−γ)Γ(2−µ+γ) z γ I −µ,γ,η f (z); −∞ ≤ µ < 0 0,z Γ(2−γ+η) µ,γ,η J0,z
where and are fractional differential and fractional integral operators [12] respectively. It is interesting to observe that
(1.9)
G. Murugusundaramoorthy, Thomas Rosy and Maslina Darus
Title Page
−µ,γ,η I0,z
Uzµ f (z) = Γ(2 − µ)z µ Dzµ f (z), = Ωµz f (z)
A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus Operators
−∞ −1, for which Vλ is known as the Bernardi operator, and δ−1 1 (c + 1)δ c t log , c > −1, δ ≥ 0 λ(t) = λ(δ) t which gives the Komatu operator. For more details see [5]. First we show that the class T R(µ, γ, η, α) is closed under Vλ (f ). Theorem 4.1. Let f ∈ T R(µ, γ, η, α). Then Vλ (f ) ∈ T R(µ, γ, η, α). Proof. By definition, we have ! Z ∞ X (c + 1)δ 1 an z n tn−1 dt Vλ (f ) = (−1)δ−1 tc (log t)δ−1 z − λ(δ) 0 n=2 "Z ! # ∞ 1 δ−1 δ X (−1) (c + 1) = lim tc (log t)δ−1 z − an z n tn−1 dt , r→0+ λ(δ) r n=2 and a simple calculation gives Vλ (f )(z) = z −
δ ∞ X c+1 n=2
c+n
A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus Operators G. Murugusundaramoorthy, Thomas Rosy and Maslina Darus
Title Page Contents
JJ J
II I
Go Back Close Quit
an z n .
Page 17 of 21 J. Ineq. Pure and Appl. Math. 6(3) Art. 86, 2005
We need to prove that (4.1)
∞ X 2n − 1 − α n=2
1−α
(2 − γ + η)n−1 (2)n−1 · (2 − γ)n−1 (2 − µ + η)n−1
c+1 c+n
δ an < 1.
On the other hand by Theorem 2.1, f ∈ T R(µ, γ, η, α) if and only if ∞ X 2n − 1 − α
1−α
n=2
Hence
c+1 c+n
·
(2 − γ + η)n−1 (2)n−1 < 1. (2 − γ)n−1 (2 − µ + η)n−1
A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus Operators
< 1. Therefore (4.1) holds and the proof is complete.
Next we provide a starlike condition for functions in T R(µ, γ, η, α) and Vλ (f ). Theorem 4.2. Let f ∈ T R(µ, γ, η, α). Then Vλ (f ) is starlike of order 0 ≤ γ < 1 in |z| < R1 where " R1 = inf n
c+n c+1
1 # n−1
δ ·
1 − γ(2n − 1 − α) φ(n) (n − γ)(1 − α)
Proof. It is sufficient to prove z(Vλ (f )(z))0 < 1 − γ. (4.2) − 1 Vλ (f )(z)
.
G. Murugusundaramoorthy, Thomas Rosy and Maslina Darus
Title Page Contents
JJ J
II I
Go Back Close Quit Page 18 of 21 J. Ineq. Pure and Appl. Math. 6(3) Art. 86, 2005
For the left hand side of (4.2) we have P∞ c+1 δ z(Vλ (f )(z))0 n=2 (1 − n)( c+n ) an z n−1 Vλ (f )(z) − 1 = 1 − P∞ ( c+1 )δ a z n−1 n n=2 c+n P∞ c+1 δ n−1 n=2 (n − 1)( c+n ) an |z| P ≤ . c+1 δ n−1 1− ∞ n=2 ( c+n ) an |z| This last expression is less than (1 − γ) since |z|n−1
