A Subclass of Uniformly Convex Functions Associated with ... - EMIS

Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 6, Issue 3, Article 86, 2005

A SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS ASSOCIATED WITH CERTAIN FRACTIONAL CALCULUS OPERATORS G. MURUGUSUNDARAMOORTHY, THOMAS ROSY, AND MASLINA DARUS D EPARTMENT OF M ATHEMATICS V ELLORE I NSTITUTE OF T ECHNOLOGY, D EEMED U NIVERSITY V ELLORE - 632014, I NDIA . [email protected] D EPARTMENT OF M ATHEMATICS M ADRAS C HRISTIAN C OLLEGE , C HENNAI - 600059, I NDIA . S CHOOL OF M ATHEMATICAL S CIENCES FACULTY OF S CIENCE AND T ECHNOLOGY, U NIVERSITI K EBANGSAAN M ALAYSIA , BANGI 43600 S ELANGOR , M ALAYSIA . [email protected]

Received 03 March, 2005; accepted 26 July, 2005 Communicated by G. Kohr

A BSTRACT. 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. Key words and phrases: Univalent, Convex, Starlike, Uniformly convex , Fractional calculus integral operator. 2000 Mathematics Subject Classification. 30C45.

1. I NTRODUCTION AND P RELIMINARIES 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. ISSN (electronic): 1443-5756 c 2005 Victoria University. All rights reserved.

062-05

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G. M URUGUSUNDARAMOORTHY , T HOMAS ROSY , AND M ASLINA DARUS

Denote by A the class of functions of the form f (z) = z +

(1.1)

∞ 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 f (z) = z −

(1.2)

∞ 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) − α ≥ β 0 − 1 , (1.3) Re 1 + 0 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) (1.4) Re − α ≥ β − 1 , f (z) f (z) 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

zn,

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 } Further suppose L(a, c)f (z) = φ(a, c; z) ∗ f (z),

for

f ∈A

where L(a, c) is called Carlson - Shaffer operator [2]. 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 (1.7)

µ,γ,η U0,z

=z+

∞ X n=2

J. Inequal. Pure and Appl. Math., 6(3) Art. 86, 2005

(2 − γ + η)n−1 (2)n−1 an z n , (2 − γ)n−1 (2 − µ + η)n−1

http://jipam.vu.edu.au/

U NIFORMLY C ONVEX F UNCTIONS

which for f (z) 6= 0 may be written as  Γ(2 − γ)Γ(2 − µ + γ) γ µ,γ,η   z J0,z f (z);   Γ(2 − γ + η) µ,γ,η (1.8) U0,z f (z) =  Γ(2 − γ)Γ(2 − µ + γ) γ −µ,γ,η   z I0,z f (z);  Γ(2 − γ + η)

3

0≤µ −1, δ ≥ 0 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 Vλ (f ) = (−1)δ−1 tc (log t)δ−1 z − an z n tn−1 dt 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

an z n .

We need to prove that δ ∞ X 2n − 1 − α (2 − γ + η)n−1 (2)n−1 c+1 (4.1) · an < 1. 1−α (2 − γ)n−1 (2 − µ + η)n−1 c + n n=2



U NIFORMLY C ONVEX F UNCTIONS

9

On the other hand by Theorem 2.1, f ∈ T R(µ, γ, η, α) if and only if ∞ X 2n − 1 − α (2 − γ + η)n−1 (2)n−1 · < 1. 1 − α (2 − γ) (2 − µ + η) n−1 n−1 n=2 Hence

c+1 c+n

< 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 1 " # n−1 δ c+n 1 − γ(2n − 1 − α) . R1 = inf · φ(n) n c+1 (n − γ)(1 − α) Proof. It is sufficient to prove z(Vλ (f )(z))0 (4.2) Vλ (f )(z) − 1 < 1 − γ. For the left hand side of (4.2) we have P∞ c+1 δ n=2 (1 − n)( c+n z(Vλ (f )(z))0 ) 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 δ (1 − γ)[2n − 1 − α] c+1 n−1 φ(n). |z| < c+n (n − γ)(1 − α) Therefore the proof is complete.

Using the fact that f is convex if and only if zf 0 is starlike, we obtain the following: Theorem 4.3. Let f ∈ T R(µ, γ, η, α). Then Vλ (f ) is convex of order 0 ≤ γ < 1 in |z| < R2 where 1 " # n−1 δ c + n (1 − γ)[2n − 1 − α] R2 = inf φ(n) . n c+1 n(n − γ)(1 − α) We omit the proof as it is easily derived. R EFERENCES [1] R. BHARATI, R. PARVATHAM AND A. SWAMINATHAN, On subclass of uniformly convex functions and corresponding clss of starlike functions, Tamkang J. of Math., 28(1) (1997), 17–33. [2] B.C. CARLSON AND S.B. SHAFFER, Starlike and prestarlike hypergrometric functions, SIAM J. Math. Anal., 15 (2002), 737–745. [3] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 56 (1991), 87–92. [4] A.W. GOODMAN, On uniformly starlike functions, J. Math. Anal. & Appl., 155 (1991), 364–370. [5] Y.C. KIM AND F. RØNNING, Integral transform of certain subclasses of analytic functions, J. Math. Anal. Appl., 258 (2001), 466–489. [6] W. MA AND D. MINDA, Uniformly convex functions, Ann. Polon. Math., 57 (1992), 165–175.



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AND

M ASLINA DARUS

[7] S. OWA, On the distribution theorem I, Kyungpook Math. J., 18 (1978), 53–59. [8] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196. [9] F. RØNNING, Integral representations for bounded starlike functions, Annal. Polon. Math., 60 (1995), 289–297. [10] A. SCHILD AND H. SILVERMAN, Convolution of univalent functions with negative coefficients, Ann. Univ. Marie Curie-Sklodowska Sect. A, 29 (1975), 99–107. [11] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 109–116. [12] H.M. SRIVASTAVA AND S. OWA, An application of the fractional derivative, Math. Japonica, 29 (1984), 383–389.

