On Classes Of Analytic Functions Involving A Linear Fractional Differential Operator

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1 International Journal of Pure and Applied Mathematical Sciences. ISSN Volume 9, Number 1 (216), pp Research India Publications On Classes Of Analytic Functions Involving A Linear Fractional Differential Operator S. Ramanathan Department of Mathematics and Statistics, Caledonian College of Engineering, Muscat, Sultanate of Oman. ramakshmi@gmail.com Abstract A new class of analytic functions is defined using a new linear multiplier fractional differential operator. Characterization property, Co-efficient bounds, distortion theorem, integral means inequalities and some interesting subordination results are obtained. AMS subject classification: 3C45, 26A33, 3C5. Keywords: Analytic functions, Hadamard product, Subordinating sequence, Fractional Calculus. 1. Introduction, Definitions and Preliminaries Let A be the class of analytic functions f(z)= z + a n z n (1.1) defined on the open unit disc U ={z : z < 1}. Let S denote the subclass of A consisting of functions that are univalent in U. The Hadamard product of two functions f(z)= z + a n z n and g(z) = z + b n z n in A is given by (f g)(z) = z + a n b n z n. (1.2)

2 2 S. Ramanathan Let the function φ(b,c; z) be given by φ(b,c; z) = z + (b) n 1 (c) n 1 z n (c =, 1, 2,... : z U) where (x) n is the Pochhammer symbol defined, in terms of the Gamma function Ɣ, by { Ɣ(x + n) 1 if n = (x) n = = Ɣ(x) x(x + 1)(x + 2)... (x+ n 1) if n N ={1, 2,,...}. Further let L(b, c)f (z) = φ(b,c; z) f(z)= z + (b) n 1 (c) n 1 a n z n, f A where L(b, c) is called Carlson-Shaffer operator [4]. Motivated by the Carlson-Shaffer operator, Murugusundaramoorthy et al. [9] defined the operator U z (µ,γ,η): A A by U z (µ, γ, η)f (z) = z + (2 γ + η) n 1 (2) n 1 a n z n, (1.3) where <µ,γ <1 and η R, which for f(z) = may be written as Ɣ(2 γ)ɣ(2 µ + γ) z γ J z µ,γ,η f(z) µ<1 Ɣ(2 γ + η) U z (µ,γ,η)= Ɣ(2 γ)ɣ(2 µ + γ) z γ Iz µ,γ,η f(z) µ< Ɣ(2 γ + η) (1.4) where J z µ,γ,η f(z)and Iz µ,γ,η f(z)are fractional differential and fractional integral operator [6] respectively. We observe that U z (µ)f (z) = Ɣ(2 µ)z µ D z µ f(z)= µ z f(z) <µ<1 (1.5) and D z µ is due to Owa [13]. U z µ is called a fractional integral operator of order µ, if <µ<and is called fractional integral operator of order µ if µ<1. Corresponding to the function (1.3), we now define the operator Dλ m (µ,γ,η): A A by Dλ (µ, γ, η)f (z) = U z(µ, γ, η)f (z) D 1 λ (µ, γ, η)f (z) = (1 λ)u z(µ, γ, η)f (z) + λz(u z (µ, γ, η)f (z)) (1.6) D m λ (µ, γ, η)f (z) = D1 λ (Dm 1 λ (µ, γ, η)f (z)) (1.7)

3 Functions Involving A Linear Fractional Differential Operator 3 If f A, then from (1.6) and (1.7) we may easily deduce that D m λ (µ, γ, η)f (z) = z + (2 γ + η) n 1 (2) n 1 [ 1 + (n 1)λ ] man z n (1.8) where <µ,γ <1, η R, m N = N {} and λ. It is interesting to note that Dλ m (µ, γ, η)f (z) is neither the generalization nor the special case of the operator recently introduced by F.M. Al-Oboudi and K.A.Al. Amoudi [2]. When µ = γ =, we get Al-Oboudi differential operator [1], when m = we get the operator introduced by Murugusundaramoorthy et al. [9], when µ = γ =, λ= 1, we get the Sălăgean differential operator [11]. Using the operator Dλ m (µ, γ, η)f (z), we now define the following class. Definition 1.1. Let Uλ m (µ,γ,η; α, β) consist of function f A satisfying the condition { z(d m Re λ (µ, γ, η)f (z)) } >α z(d m λ (µ, γ, η)f (z)) (µ, γ, η)f (z) Dλ m(µ, γ, η)f (z) 1 + β (1.9) D m λ The family Uλ m (µ,γ,η; α, β) is of special interest for it contains many well known as well as many new classes of analytic univalent functions. For Uλ (µ,γ,η; α, ) is the class introduced by Murugusundaramoorthy et al. [9]. For Uλ (,,η; α, β) and Uλ (1, 1,η; α, β), we obtain the well known family of uniformly starlke function [3] and uniformly convex functions [3] respectively. For Uλ 1 (,,η;,β), we obtain the family of λ-pacasu convex function of order α. Further we observe that many subclasses of analytic are the special cases of Uλ m (µ,γ,η; α, β) (see [5, 7, 1, 15]). Let T denote the subclass of A introduced and studied by Silverman [12], consisting of functions of the form f(z)= z a n z n a n. (1.1) Now let us write TUλ m (µ,γ,η; α, β) = U λ m (µ,γ,η; α, β) T. 2. Characterization Property Theorem 2.1. A function f TUλ m (µ,γ,η; α, β) if and only if ( )[ ] m (2 γ + η) n 1 (2) n 1 n(1+α) (α+β) 1+(n 1)λ a n (1 β), (2.1) where β<1, α, <µ,γ <1 and η R. Proof. It suffices to show that α z(d m { λ (µ, γ, η)f (z)) Dλ m(µ, γ, η)f (z) 1 z(d m Re λ (µ, γ, η)f (z)) } Dλ m(µ, γ, η)f (z) 1 1 β.

4 4 S. Ramanathan We have α z(d m { λ (µ, γ, η)f (z)) Dλ m(µ, γ, η)f (z) 1 z(d m Re λ (µ, γ, η)f (z)) } Dλ m(µ, γ, η)f (z) 1 (1 + α) z(d m λ (µ, γ, η)f (z)) Dλ m(µ, γ, η)f (z) 1 (1 + α) (n 1) [ 1 + (n 1)λ ] m (2 γ +η) n 1 (2) n 1 (2 γ) n 1 (2 µ+η) n 1 a n z n 1 1 [ ] m 1 + (n 1)λ (2 γ +η)n 1 (2) n 1 (2 γ) n 1 (2 µ+η) n 1 a n z n 1 (1 + α) (n 1) [ 1 + (n 1)λ ] m (2 γ +η) n 1 (2) n 1 (2 γ) n 1 (2 µ+η) n 1 a n 1 [ ] m 1 + (n 1)λ (2 γ +η)n 1. (2) n 1 (2 γ) n 1 (2 µ+η) n 1 a n The last expression is bounded by 1 β if ( n(1 + α) (α + β) )[ 1 + (n 1)λ ] m (2 γ + η) n 1 (2) n 1 a n (1 β) which completes the proof of the theorem. Theorem 2.2. If the function f TUλ m (µ,γ,η; α, β), then for z = r<1 a n (1 β) ( n(1 + α) (α + β) ) (2 γ + η)n 1 (2) n 1 [ 1 + (n 1)λ ] m. (2.2) Next we consider the growth and distortion theorem for the class TUλ m (µ,γ,η; α, β). We shall omit the proof as the techniques are similar to various other papers. Theorem 2.3. Let f(z) be given by (1.1). z = r<1wehave If f TUλ m (µ,γ,η; α, β), then for (1 β)(2 γ)(2 µ + η) r 2(2 γ + η) (2 + α β) [ 1 + λ ] m r2 f(z) (1 β)(2 γ)(2 µ + η) r + 2 (2 γ + η) (2 + α β) [ 1 + λ ] m r2 (2.3) and (1 β)(2 γ)(2 µ + η) 1 (2 γ + η) (2 + α β) [ 1 + λ ] m r f (1 β)(2 γ)(2 µ + η) (z) 1 (2 γ + η) (2 + α β) [ 1 + λ ] m r. (2.4)

5 Functions Involving A Linear Fractional Differential Operator 5 The results (2.3) and (2.4) are sharp for the function f(z)given by f(z)= z (1 β)(2 γ)(2 µ + η) 2(2 γ + η) (2 + α β) [ 1 + λ ] m z2. (2.5) By taking different choices of µ, γ, η; α, β, λ and m in the above theorems, we have the characterization, distortion theorem, growth theorem and co-efficient estimates for various subclasses studied earlier by several researchers (see for example [3, 7, 9, 12, 15]). 3. Integral Means Inequalities for the class TU m λ (µ,γ,η; α, β) An analytic function g is said to be subordinate to an analytic function f if g(z) = f (w(z)), z U for some analytic function w with w(z) z. We need the following subordination theorem of Littlewood [8] in our investigation. Lemma 3.1. If the functions f and g are analytic in U with g(z) f(z), then, for τ> and z = re iθ ( <r<1), g(re iθ ) τ dθ f(re iθ ) τ dθ. Theorem 3.2. Let τ>.if f TUλ m (µ,γ,η; α, β) and suppose that (1 β) f 2 (z) = z (2 + α β) 2 (µ,γ,η) z2, where n (µ,γ,η)= [ 1 + (n 1)λ ] m (2 γ + η) n 1 (2) n 1, then for z = re iθ and <r<1, Proof. In view of the functions f(re iθ ) τ dθ f(z)= z f 2 (re iθ ) τ dθ (3.1) a n z n, a n, for n 2, (3.2) (1 β) f 2 (z) = z (2 + α β) 2 (µ,γ,η) z2, then we must show that 1 a n z n 1 τ dθ 1 (1 β) (2 + α β) 2 (µ,γ,η) z τ dθ. (3.3)

6 6 S. Ramanathan By Lemma (3.4), it suffices to show that Set 1 1 a n z n 1 (1 β) 1 z. (3.4) (2 + α β) 2 (µ,γ,η) From (3.5) and Theorem (2.1), we obtain a n z n 1 (1 β) = 1 w(z). (3.5) (2 + α β) 2 (µ,γ,η) w(z) = (2 + α + β) 2 (µ,γ,η) (1 β) a n z n 1 ( ) n(1 + α) (α + β) z n 1 n (µ,γ,η) a n z n 1 z. (1 β) This means that the subordination (3.4) holds true, therefore the theorem is proved. Theorem 3.3. Let τ>. If f TUλ m (µ,γ,η; α, β) and f 2 (z) = z (1 β) (2 + α β) 2 (µ,γ,η) z2, then for z = re iθ and <r<1, f (re iθ ) τ dθ Proof. By applying Lemma (3.4), it would suffice to show that 1 f 2 (reiθ ) τ dθ (3.6) na n z n 1 2(1 β) 1 z. (3.7) (2 + α β) 2 (µ,γ,η) This follows because ( ) n(1 + α) (α + β) n (µ,γ,η) w(z) = a n z n 1 (1 β) ( ) n(1 + α) (α + β) n (µ,γ,η) z a n (1 β) z. n=m

7 Functions Involving A Linear Fractional Differential Operator 7 4. Subordination Results For the Class U m λ (µ,γ,η; α, β) Definition 4.1. A sequence {b n } n=1 of complex numbers is called a subordinating factor sequence if, whenever f(z) is analytic, univalent and convex in U, wehavethe subordination given by b n a n z n f(z) (z U, a 1 = 1). (4.1) n=1 Lemma 4.2. [17] The sequence {b n } n=1 is a subordinating factor sequence if and only if { } Re b n z n > (z U). (4.2) n=1 Appealing to Theorem (2.1), we have the following lemma. Lemma 4.3. If ( )[ ] m n(1 + α) (α + β) 1 + (n 1)λ then f(z) Uλ m (µ,γ,η; α, β). (2 γ + η) n 1 (2) n 1 a n (1 β), (4.3) For convenience, we shall henceforth denote σ n (γ,µ,η,α,β)= ( n(1 + α) (α + β) )[ 1 + (n 1)λ ] m (2 γ + η) n 1 (2) n 1. (4.4) Let Ũλ m (µ,γ,η; α, β) denote the class of functions f(z) A whose coefficients satisfy the conditions (4.3). We note that Ũλ m (µ,γ,η; α, β) U λ m (µ,γ,η; α, β). Theorem 4.4. Let the function f(z)defined by (1.1) be in the class Ũλ m (µ,γ,η; α, β) where 1 β<1; α. Also let C denote the familiar class of functions f(z) A which are also univalent and convex in U. Then 2 [ ](f g)(z) g(z) (z U; g C), (4.5) 1 β + and Re(f (z)) > 1 β + σ 2(γ,µ,η,α,β) (z U). (4.6)

8 8 S. Ramanathan The constant is the best estimate. 2 [ 1 β + ] Proof. Let f(z) Ũλ m (µ,γ,η; α, β) and let g(z) = z + b n z n C. Then 2 [ ](f g)(z) 1 β + ( = 2 [ 1 β + ] z + a n b n z ). n Thus, by Definition (4.7), the assertion of the theorem will hold if the sequence { } σ n (γ,µ,η,α,β) 2 [ 1 β + σ n (γ,µ,η,α,β) ]a n n=1 is a subordinating factor sequence, with a 1 = 1. In view of Lemma (4.8), this will be true if and only if { Re Now n=1 { Re 1 + { = Re } 2 [ 1 β + ]a nz n > (z U). (4.7) 1 β } a n z n n=1 1 β + a 1z 1 } σ n (γ,µ,η,α,β)a n z n + 1 β + { 1 1 β + r 1 1 β + σ n (γ,µ,η,α,β)a n r n }. Since σ n (γ,µ,η,α,β)is an increasing function of n (n 2) { 1 1 β + r 1 } σ n (γ,µ,η,α,β)a n r n 1 β +

9 Functions Involving A Linear Fractional Differential Operator 9 > 1 1 β + r 1 β 1 β + r>. Thus (4.7) holds true in U. This proves the inequality (4.5). The inequality (4.6) follows by taking the convex function g(z) = z 1 z = z+ z n in (4.5). To prove the sharpness of the constant 2 [ 1 γ + ], we consider given by f (z) = z Thus from (4.5), we have f (z) Ũλ m (µ,γ,η; α, β) 1 β z2 ( 1 β<1; γ ). 2 [ 1 β + ]f (z) z 1 z. (4.8) It can be easily verified that { ( )} min Re 2 [ 1 β + ]f (z) = 1 2 This shows that the constant 2 [ 1 β + ] (z U), is best possible. By taking k = α = µ = γ = and λ = 1 in Theorem (4.1), we get Corollary 4.5. [14] Let f(z)be defined by (1.1) be in the class S m (β) and satisfy the ( coefficient inequality n m+1 βn m) a n 1 β, β<1. Also let C denote the familiar class of functions f(z) A which are univalent and convex in U. Then and 2 m β2 m 1 (1 β) + (2 m+1 β2 m (f g)(z) g(z) (z U; g C), (4.9) ) Re(f (z)) > (1 β) + (2m+1 β2 m ) (2 m+1 β2 m. (4.1) )

10 1 S. Ramanathan The constant is the best estimate. 2 m β2 m 1 (1 β) + (2 m+1 β2 m ) Corollary 4.6. [16] Let the function f(z)be defined by (1.1) be starlike in U and satisfy the condition n a n 1, then and 1 (f g)(z) g(z) (z U; g C) (4.11) 3 Re(f (z)) > 3, (z U). (4.12) 2 Corollary 4.7. L et the function f(z) be defined by (1.1) be convex in U and satisfy the condition n 2 a n 1, then and 2 (f g)(z) g(z) (z U; g C) (4.13) 5 Re(f (z)) > 5, (z U). (4.14) 4 References [1] F. M. Al-Oboudi, On univalent functions defined by a generalized Sălăgean operator. Int. J. Math. Math. Sci , (24), [2] F. M. Al-Oboudi, K.A.Al-Omoudi, On Classes of analytic functions related to conic domains. J. Math. Anal. Appl. 339, (28), [3] R. Bharati, R. Parvatham, A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions. Tamkang J. Math. 28, (1997), no 1, [4] B. C. Carlson, D. B. Shaffer, Starlike and prestarlike hypergeometric functions. SIAM J. Math. Anal. 15, (1984), no 4, [5] P. L. Duren, Univalent functions. Springer, New York, (1983). [6] H. M. Srivastava, S. Owa, An application of the fractional derivative. Math. Japon. 29, (1984), no 3, [7] S. Kanas, H. M. Srivastava, Linear operators associated with k-uniformly convex functions. Integral Transform. Spec. Funct. 9, (2), no 2,

11 Functions Involving A Linear Fractional Differential Operator 11 [8] J. E. Littlewood, On inequalities in the theory of functions. Proceedings of the London Mathematical Society 23, (1925), no 1, [9] G. Murugusundaramoorthy, T. Rosy, M. Darus, A subclass of uniformly convex functions associated with certain fractional calculus operators. JIPAM. J. Inequal. Pure Appl. Math. 6, (25), no 3, Article 86, Article 86, 1 pp. (electronic). [1] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions. Proc. Amer. Math. Soc. 118, (1993), no 1, [11] G. Ş. Sălăgean, Subclasses of univalent functions. Complex analysis fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981) , Lecture Notes in Math., 113, Springer, Berlin. [12] H. Silverman, Univalent functions with negative coefficients. Proc. Amer. Math. Soc. 51, (1975), no 1, [13] S. Owa, On the distortion theorems I. Kyungpook Math. J. 18, (1973), no 1, [14] Sevtap Sümer Eker, Bilal Şeker, Shigeyoshi Owa, On subordination result associated with certain subclass of analytic functions involving Salagean operator. Journal of Inequalities and Applications vol. 27, Article ID 48294, 6 pages, (27). doi:1.1155/27/48294 [15] S. Shams, S. R. Kulkarni, J. M. Jahangiri, Classes of Ruscheweyh-type analytic univalent functions. Southwest J. Pure Appl. Math., (23), no 2, (electronic). [16] S. Singh, A subordination theorem for spirallike functions. Int. J. Math. Math. Sci. 24, (2), no 7, [17] H. S. Wilf, Subordinating factor sequences for convex maps of the unit circle. Proc. Amer. Math. Soc. 12, (1961),

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