International Mathematical Forum, Vol. 6, 211, no. 52, 2573-2586 Suclasses of Analytic Functions Involving the Hurwitz-Lerch Zeta Function Shigeyoshi Owa Department of Mathematics Kinki University Higashi-Osaka, Osaka 577-852, Japan owa@math.kindai.ac.jp G. Murugusundaramoorthy and K. Uma School of Advanced Science, VIT University Vellore-63214, India gmsmoorthy@yahoo.com Astract. In this paper, we introduce a generalized class of starlike functions and otain the suordination results for various suclasses of starlike functions. Further, we otain the integral means inequalities for various suclasses of starlike functions. Some interesting consequences of our results are also pointed out. Mathematics Suject Classification: 3C45, 3C8 Keywords: Univalent, starlike, convex, suordinating factor sequence, integral means, Hadamard product, Hurwitz-Lerch Zeta function
2574 Shigeyoshi Owa, G. Murugusundaramoorthy and K. Uma 1. Introduction and Preliminaries Let A denote the class of functions of the form f(z) =z + a n z n (1.1) which are analytic in the open disc U = z : z < 1. Also let T e a suclass of A consisting of functions of the form f(z) =z a n z n, z U, (1.2) introduced and studied y Silverman [17]. For functions φ Agiven y φ(z) =z + φ n z n and ψ Agiven y ψ(z) =z + ψ n z n, we define the Hadamard product (or Convolution ) of φ and ψ y (φ ψ)(z) =z + φ n ψ n z n, z U. (1.3) The following we recall a general Hurwitz-Lerch Zeta function Φ(z, s, a) defined y (cf., e.g., [22], p. 121 et sep.]) Φ(z, s, a) := n= z n (n + a) s (1.4) (a C \ Z ; s C, when z < 1; R(s) > 1 when z = 1) where, as usual, Z := Z \N, (Z :=, ±1, ±2, ±3,...); N := 1, 2, 3,... Several interesting properties and characteristics of the Hurwitz-Lerch Zeta function Φ(z, s, a) can e found in the recent investigations y Choi and Srivastava [4], Ferreira and Lopez [5], Garg et al. [8], Lin and Srivastava [13], Lin et al. [14], and others. In 27, Srivastava and Attiya [21] (see also Raducanu and Srivastava [16], and Prajapat and Goyal [15]) introduced and investigated the linear operator J μ, : A Adefined, in terms of the Hadamard product (or convolution), y (z U; C \Z ; μ C; f A), where, for convenience, J μ, f(z) =G,μ f(z) (1.5) G μ, (z) :=(1+) μ [Φ(z, μ, ) μ ] (z U). (1.6)
Suclasses of analytic functions 2575 It is easy to oserve from (1.5) and (1.6) that, for f(z) of the form(1.1),we have J μ, f(z) =z + C n (, μ)a n z n (1.7) where C n (, μ) = ( ) 1+ μ (1.8) n + and (throughout this paper unless otherwise mentioned) the parameters μ, as C \Z ; μ C. are constrained and various choices of μ we get different operators as listed elow. J f(z) :=f(z) =z + a n z n, (1.9) J 1 f(z) := z f(t) t dt := Λf(z) =z + ( ) 1 a n z n, (1.1) n J 1 ν f(z) := 1+ν z ν z t 1 ν f(t)dt := z + ( ) 1+ν a n z n, = F ν fz), (ν > 1), (1.11) n + ν J σ 1 f(z) :=z + ( ) 2 σ a n z n = I σ f(z)(σ >), (1.12) n +1 where Λf and F ν are the integral operators introduced y Alexander [1] and Bernardi [3], respectively, and I σ (f) is the Jung-Kim-Srivastava integral operator [9] closely related to some multiplier transformation studied y Flett [6]. In this paper, y making use of the operatorj μ, we introduced a new suclass of analytic functions with negative coefficients and discuss some interesting properties of this generalized function class.
2576 Shigeyoshi Owa, G. Murugusundaramoorthy and K. Uma For λ<1, γ<1and k, we let Jμ (λ, γ, k) e the suclass of A consisting of functions of the form (1.1) and satisfying the analytic criterion where J μ Re z(j μ f(z)) (1 λ)j μ f(z)+λz(j μ γ f(z)) >k z(j μ f(z)) (1 λ)j μ f(z)+λz(j μ μ f(z) is given y (1.5). We further let T J As illustrations, we present some examples. Example 1. If μ =and = 1 f(z)), z U, (1.13) μ (λ, γ, k) =J (λ, γ, k) T. J (λ, γ, k) S(λ, γ, k) zf (z) := f A: Re (1 λ)f(z)+λzf (z) γ >k zf (z) (1 λ)f(z)+λzf (z) 1, z U. Further T S(λ, γ, k) =S(λ, γ, k) T, where T is given y (1.2). Example 2. If μ =1and = ν with ν> 1, then Jν 1 (λ, γ, k) B ν(λ, γ, k) ( z(j ν f(z)) ) := f A: Re (1 λ)j ν f(z)+λz(j ν f(z)) γ >k z(j ν f(z)) (1 λ)j ν f(z)+λz(j ν f(z)) 1, z U, where J ν is a Bernardi operator [3] given y (1.11). Note that the operator J 1 was studied earlier y Liera [1] and Livingston [12]. Further, TB ν (λ, γ, k) =B ν (λ, γ, k) T, where T is given y (1.2). Example 3. If μ = σ and =1with σ>, then J1 σ (λ, γ, k) I σ (λ, γ, k) ( z(i σ f(z)) ) := f A: Re (1 λ)i σ f(z)+λz(i σ f(z)) γ >k z(i σ f(z)) (1 λ)i σ f(z)+λz(i σ f(z)) 1, z U, where I σ is the Jung-Kim-Srivastava integral operator [9] given y (1.12). Motivated y earlier works of [2, 18],[23] in this paper, we investigate certain characteristic properties and otain the suordination results for the class of functions f J μ (λ, γ, k) and integral means results for the class of functions f T J μ (λ, γ, k). We state some interesting results for functions in those classes defined in Examples 1 to 4.
Suclasses of analytic functions 2577 2. Basic Properties In this section we otain the characterization properties for the classes J μ (λ, γ, k) and T J μ (λ, γ, k). Theorem 2.1. A function f(z) of the form (1.1) is in J μ (λ, γ, k) if [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ) a n 1 γ, (2.1) where λ<1, γ<1, k, and C n (, μ) is given y (1.8). Proof. It suffices to show that k z(j μ f(z)) (1 λ)j μ f(z)+λz(j μ 1 f(z)) z(j μ Re f(z)) (1 λ)j μ f(z)+λz(j μ 1 f(z)) We have. k z(j μ f(z)) (1 λ)j μ f(z)+λz(j μ 1 f(z)) z(j μ Re f(z)) (1 λ)j μ f(z)+λz(j μ 1 f(z)) (1 + k) z(j μ f(z)) (1 λ)j μ f(z)+λz(j μ 1 f(z)) (1 + k) (n 1 nλ + λ)c n (, μ) a n z n 1 1 (1 + nλ λ)c n (, μ) a n z n 1 (1 + k) (n 1 nλ + λ)c n (, μ) a n < 1. (1 + nλ λ)c n (, μ) a n The last expression is ounded aove y () if [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ) a n 1 γ and the proof is complete.
2578 Shigeyoshi Owa, G. Murugusundaramoorthy and K. Uma Various choices of and μ and in the view of Examples 1 and 2, we state the following corollaries. Corollary 2.1. A function f(z) of the form (1.1) is in S(λ, γ, k) if [n(1 + k) (γ + k)(1 + nλ λ)] a n 1 γ, (2.2) where λ<1, γ<1 and k. Corollary 2.2. A function f(z) of the form (1.1) is in B ν (λ, γ, k) if ( ) ν +1 [n(1 + k) (γ + k)(1 + nλ λ)] a n 1 γ, (2.3) ν + n where λ<1, γ<1, k and ν> 1. Theorem 2.2. Let λ<1, γ<1, k, then a function f of the form (1.2) to e in the class T J μ (λ, γ, k) if and only if [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ) a n 1 γ, (2.4) where C n (, μ) are given y (1.8). Proof. In view of Theorem 2.1, we need only to prove the necessity. If f T J μ (λ, γ, k) and z is real then 1 nc n (, μ) a n z n 1 (n 1 nλ + λ)c n (, μ) a n z n 1 Re 1 γ >k [1 + nλ λ]c n (, μ) a n z n 1 1. [1 + nλ λ]c n (, μ) a n z n 1 Letting z 1 along the real axis, we otain the desired inequality [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ) a n 1 γ, where λ<1, γ<1, k, and C n (, μ) are given y (1.8). Corollary 2.3. If f T J μ (λ, γ, k), then a n, λ<1, γ<1,k, (2.5) [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ)
Suclasses of analytic functions 2579 where C n (, μ) are given y (1.8). Equality holds for the function f(z) =z Theorem 2.3. (Extreme Points) Let [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ) zn. f 1 (z) =z f n (z) =z and [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ) zn, n 2, (2.6) for γ < 1, λ<1, k, C n (, μ) are given y (1.8). T J μ (λ, γ, k) if and only if it can e expressed in the form where ω n and ω n =1. f(z) = Then f(z) is in the class ω n f n (z), (2.7) Proof. Suppose f(z) can e written as in (2.7). Then Now, f(z) = z ω n [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ) = ω n =1 ω 1 1. Thus f T J μ [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ) zn. ω n [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ) μ (λ, γ, k). Conversely, let us have f T J (λ, γ, k). Then y using (2.5), we set ω n = [n(1 + k) (γ + k)(1 + nλ λ)]c n(, μ) a n, n 2 and ω 1 =1 ω n. Then we have f(z) = ω n f n (z) and hence this completes the proof of Theorem 2.3. Remark 2.1. By suitaly specializing the various parameters involved in Theorem 2.2 and Theorem 2.3, we can state the corresponding results for the new suclasses defined in Example 1 to 3 and also for many relatively more familiar function classes.
258 Shigeyoshi Owa, G. Murugusundaramoorthy and K. Uma 3. Suordination Results In this section we otain suordination results for the new class Wm l (λ, γ, k) Definition 3.1. (Suordination Principle) For analytic functions g and h with g() = h(), g is said to e suordinate to h, denoted y g h, if there exists an analytic function w such that w() =, w(z) < 1 and g(z) =h(w(z)), for all z U. Definition 3.2. (Suordinating Factor Sequence) A sequence n of complex numers is said to e a suordinating sequence if, whenever f(z) = a n z n,a 1 =1is regular, univalent and convex in U, we have n a n z n f(z), z U. (3.1) Lemma 3.1. [23] The sequence n is a suordinating factor sequence if and only if Re 1+2 n z n >, z U. (3.2) Theorem 3.1. Let f W l m(λ, γ, k) and g(z) e any function in the usual class of convex functions C, then (2 + k γ λ(k + γ))c 2 (, s) (f g)(z) g(z) (3.3) 2[ +(2+k γ λ(k + γ))c 2 (, μ)] where γ<1; k and λ<1, with and C 2 (, μ) = ( ) 1+ μ (3.4) 2+ Re f(z) > [ +(2+k γ λ(k + γ))c 2(, μ)], z U. (3.5) (2 + k γ λ(k + γ))c 2 (, μ) (2+k γ λ(k+γ))c The constant factor 2 (,μ) 2[1 γ+(2+k γ λ(k+γ))c 2 (,μ)] in (3.3) cannot e replaced y a larger numer. Proof. Let f Wm(λ, l γ, k) and suppose that g(z) =z + n z n C. Then (2 + k γ λ(k + γ))c 2 (, μ) (f g)(z) 2[ +(2+k γ λ(k + γ))c 2 (, μ)] = (2 + k γ λ(k + γ))c 2 (, μ) 2[ +(2+k γ λ(k + γ))c 2 (, μ)] ( ) z + n a n z n. (3.6)
Suclasses of analytic functions 2581 Thus, y Definition 3.2, the suordination result holds true if (2 + k γ λ(k + γ))c 2 (, μ) 2[ +(2+k γ λ(k + γ))c 2 (, μ)] is a suordinating factor sequence, with a 1 = 1. In view of Lemma 3.1, this is equivalent to the following inequality (2 + k γ λ(k + γ))c 2 (, μ) Re 1+ [ +(2+k γ λ(k + γ))c 2 (, μ)] a nz n >, z U. (3.7) By noting the fact that (n(1+k) (γ+k)(1+nλ λ))cn(,μ) (1 γ) particular (2 + k γ λ(k + γ))c 2 (, μ) () therefore, for z = r<1, we have Re =Re 1 1+ 1+ (2 + k γ λ(k + γ))c 2 (, μ) [ +(2+k γ λ(k + γ))c 2 (, μ)] is increasing function for n 2 and in (n(1 + k) (γ + k)(1 + nλ λ))c n(, μ), n 2, () a n z n (2 + k γ λ(k + γ))c 2 (, μ) [ +(2+k γ λ(k + γ))c 2 (, μ)] z + (2 + k γ λ(k + γ))c 2 (, μ) [ +(2+k γ λ(k + γ))c 2 (, μ)] r 1 [ +(2+k γ λ(k + γ))c 2 (, μ)] (2 + k γ λ(k + γ))c 2 (, μ)a n z n [ +(2+k γ λ(k + γ))c 2 (, μ)] [n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ)a n r n (2 + k γ λ(k + γ))c 2 (, μ) 1 [ +(2+k γ λ(k + γ))c 2 (, μ)] r [ +(2+k γ λ(k + γ))c 2 (, μ)] r >, z = r<1, where we have also made use of the assertion (2.1) of Theorem 2.1. This evidently proves the inequality (3.7) and hence also the suordination result (3.3) asserted y Theorem 3.1. The inequality (3.5) follows from (3.3) y taking g(z) = z 1 z = z + z n C. Next we consider the function F (z) :=z (2 + k γ λ(k + γ))c 2 (, μ) z2
2582 Shigeyoshi Owa, G. Murugusundaramoorthy and K. Uma where γ<1, k, λ<1 and σ 2 (α 1 ) is given y (3.4). Clearly F Wm l (λ, γ, k). For this function (3.3)ecomes (2 + k γ λ(k + γ))c 2 (, μ) 2[ +(2+k γ λ(k + γ))c 2 (, μ)] F (z) z 1 z. It is easily verified that ( ) (2 + k γ λ(k + γ))c 2 (, μ) min Re 2[ +(2+k γ λ(k + γ))c 2 (, μ)] F (z) = 1 2, z U. This shows that the constant (2+k γ λ(k+γ))c 2 (,μ) 2[1 γ+(2+k γ λ(k+γ))c 2 (,μ)] cannot e replaced y any larger one. By taking μ, and in view of the Examples 1 to 2 in Section 1, we state the following corollaries for the suclasses defined in those examples. Corollary 3.1. If f S ( λ, γ, k), then [2 + k γ λ(γ + k)] (f g)(z) g(z), (3.8) 2[3 + k 2γ λ(k + γ)] where γ<1, λ<1, k, g C and [3 + k 2γ λ(k + γ)] Ref(z) > [2 + k γ λ(k + γ)], z U. The constant factor in (3.8) cannot e replaced y a larger one. [2 + k γ λ(γ + k)] 2[3 + k 2γ λ(k + γ)] Corollary 3.2. If f B ν (λ, γ, k), then (ν + 1)[2 + k γ λ(k + γ)] (f g)(z) g(z), (3.9) 2[(ν + 2)()+(ν + 1)[2 + k γ λ(k + γ)]] where γ<1, λ<1, k, ν > 1, g C and Ref(z) > The constant factor in (3.9) cannot e replaced y a larger one. [(ν + 2)()+(ν + 1)(2 + k γ λ(k + γ))], z U. (ν + 1)(2 + k γ λ(k + γ)) (ν + 1)[2 + k γ λ(k + γ)] 2[(ν + 2)()+(ν + 1)[2 + k γ λ(k + γ)]]
Suclasses of analytic functions 2583 Remark 3.1. We oserve that Corollary 3.1, yields the results otained y Frasin [7] and Singh [2] for the special values of λ, γ and k. 4. Integral Means Inequalities. In this section, we otain integral means inequalities for the functions in the family T J μ (λ, γ, k) Lemma 4.1. [11] If the functions f and g are analytic in U with g f, then for η>, and <r<1, g(re iθ ) η dθ f(re iθ ) η dθ. (4.1) In [17], Silverman found that the function f 2 (z) =z z2 2 is often extremal over the family T and applied this function to resolve his integral means inequality, conjectured in [18] and settled in [19], that f(re iθ ) η dθ f 2 (re iθ ) η dθ, for all f T, η > and <r<1. In [19], Silverman also proved his conjecture for the suclasses T (γ) and C(γ) oft. Applying Lemma 4.1, Theorem 2.2 and Theorem 2.3, we prove the following result. Theorem 4.1. Suppose f T J μ (λ, γ, k), η>, λ<1, γ<1, k and f 2(z) is defined y where f 2 (z) =z Φ(λ, γ, k, 2) z2, Φ(λ, γ, k, 2) = [2 + k γ λ(k + γ)]c 2 (, μ) (4.2) and C 2 (, μ) is given y (3.4). Then for z = re iθ, <r<1, we have f(z) η dθ f 2 (z) η dθ. (4.3)
2584 Shigeyoshi Owa, G. Murugusundaramoorthy and K. Uma Proof. For f(z) =z a n z n, (4.3) is equivalent to proving that 1 η a n z n 1 dθ () 1 Φ(λ, γ, k, 2) z By Lemma 4.1, it suffices to show that 1 a n z n 1 1 Φ(λ, γ, k, 2) z. Setting and using (2.4), we otain 1 a n z n 1 =1 w(z), (4.4) Φ(λ, γ, k, 2) w(z) = z z, Φ(λ, γ, k, n) a n z n 1 Φ(λ, γ, k, n) a n where Φ(λ, γ, k, n) =[n(1 + k) (γ + k)(1 + nλ λ)]c n (, μ),. This completes the proof y Theorem 4.1. Remark 4.1. We oserve that for λ =, if μ =the various results presented in this paper would provide interesting extensions and generalizations of those considered earlier for simpler and familiar function classes studied in the literature.the details involved in the derivations of such specializations of the results presented in this paper are fairly straight- forward. η dθ. References [1] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math., 17(1915), 1222. [2] M. K. Aouf and G. Murugusundaramoorthy, On a suclass of uniformly convex functions defined y the Dziok- Srivastava operator, Australian J. Math. Analysis and Appl., 5(1) Art.3 (28). [3] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135(1969), 429 446. [4] J. Choi and H. M. Srivastava, Certain families of series associated with the Hurwitz-Lerch Zeta function, Appl. Math. Comput., 17 (25), 399-49.
Suclasses of analytic functions 2585 [5] C. Ferreira and J. L. Lopez, Asymptotic expansions of the Hurwitz-Lerch Zeta function, J. Math. Anal. Appl., 298(24), 21-224. [6] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl., 38(1972), 746765 [7] B. A. Frasin, Suordination results for a class of analytic functions defined y a linear operator, J. Ineq. Pure and Appl. Math., 7, 4 (134) (26), 1 7. [8] M. Garg, K. Jain and H. M. Srivastava, Some relationships etween the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions, Integral Transform. Spec. Funct., 17(26), 83-815. [9] I. B. Jung, Y. C. Kim AND H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl., 176(1993), 138147. [1] R. J. Liera, Some classes of regular univalent functions, Proc. Amer. Math. Soc., 16(1965), 755 758. [11] J. E. Littlewood, On inequalities in theory of functions, Proc. London Math. Soc., 23(1925), 481 519. [12] A. E. Livingston, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc., 17 (1966), 352 357. [13] S.-D. Lin and H. M. Srivastava, Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput., 154(24), 725-733. [14] S.-D. Lin, H. M. Srivastava and P.-Y. Wang, Some espansion formulas for a class of generalized Hurwitz-Lerch Zeta functions, Integral Transform. Spec. Funct., 17(26), 817-827. [15] J. K. Prajapat and S. P. Goyal, Applications of Srivastava-Attiya operator to the classes of strongly starlike and strongly convex functions, J. Math. Inequal., 3(29), 129-137. [16] D. Raducanu and H. M. Srivastava, A new class of analytic functions defined y means of a convolution operator involving the Hurwitz-Lerch Zeta function, Integral Transform. Spec. Funct., 18(27), 933-943. [17] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 19 116. [18] H. Silverman, A survey with open prolems on univalent functions whose coefficients are negative, Rocky Mt. J. Math., 21(1991), 199 1125. [19] H. Silverman, Integral means for univalent functions with negative coefficients, Houston J. Math., 23(1997), 169 174. [2] S. Singh, A suordination theorem for spirallike functions, Internat. J. Math. and Math. Sci., 24 (7) (2), 433 435. [21] H. M. Srivastava and A. A. Attiya, An integral operator associated with the Hurwitz-Lerch Zeta function and differential suordination, Integral Transform. Spec. Funct., 18(27), 27 216. [22] H. M. Srivastava and J. Choi, Series associated with the Zeta and related functions, Dordrecht, Boston, London: Kluwer Academic Pulishers, 21. [23] H. S. Wilf, Suordinating factor sequence for convex maps of the unit circle, Proc. Amer. Math. Soc., 12(1961), 689 693.
2586 Shigeyoshi Owa, G. Murugusundaramoorthy and K. Uma Received: April, 211