Neighborhood and partial sums results on the class of starlike functions involving Dziok-Srivastava operator

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1 Stud. Univ. Babeş-Bolyai Math. 58(23), No. 2, 7 8 Neighborhood and partial sums results on the class of starlike functions involving Dziok-Srivastava operator K. Vijaya and K. Deepa Abstract. In this paper, we introduce a new subclasses of univalent functions defined in the open unit disc involving Dziok-Srivastava Operator. The results on partial sums, integral means and neighborhood results are discussed. Mathematics Subject Classification (2): 3C45. Keywords: Univalent, starlike and convex functions, coefficient bounds, neighborhood results, partial sums.. Introduction Denote by A the class of functions of the form = z + a n z n (.) which are analytic in the open unit disc U = {z : z C and z < }. Further, by S we shall denote the class of all functions in A which are normalized by f() = = f () and univalent in U. Some of the important and well-investigated subclasses of the univalent function class S include (for example) the class S (α) of starlike functions of order α( α < ) if R ( functions of order α( α < ) if R ( zf (z) ) + zf (z) f (z) > α and the class K(α) of convex ) > α in U. It readily follows that f K(α) zf S (α). Denote by T the subclass of S consisting of functions of the form = z a n z n, a n, z U (.2) studied extensively by Silverman [].

2 72 K. Vijaya and K. Deepa For positive real values of α,..., α l and β,..., β m (β j,,... ; j =, 2,..., m) the generalized hypergeometric function l F m (z) is defined by (α ) n... (α l ) n z n lf m (z) l F m (α,... α l ; β,..., β m ; z) := (.3) (β ) n... (β m ) n n! n= (l m + ; l, m N := N {}; z U) where N denotes the set of all positive integers and (λ) k is the Pochhammer symbol defined by {, n = (λ) n = (.4) λ(λ + )(λ + 2)... (λ + n ), n N. The notation l F m is quite useful for representing many well-known functions such as the exponential, the Binomial, the Bessel, the Laguerre polynomial and others; for example see [3]. Let H(α,... α l ; β,..., β m ) : A A be a linear operator defined by H(α,... α l ; β,..., β m ) := z l F m (α, α 2,... α l ; β, β 2..., β m ; z) = z + Γ n a n z n (.5) where Γ n = (α ) n... (α l ) n (β ) n... (β m ) n (n )!, (.6) unless otherwise stated and the Hadamard product (or convolution) of two functions f, g A where of the form (.) and g(z) be given by g(z) = z + b n z n then g(z) = (f g)(z) = z + a n b n z n, z U. For simplicity, we can use a shorter notation Hm[α l ] for H(α,... α l ; β,..., β m ) in the sequel. The linear operator Hm[α l ] is called Dziok-Srivastava operator [3] (see [6, 8]), includes (as its special cases) various other linear operators introduced and studied by Carlson and Shaffer [2], Ruscheweyh [9] and Owa-Srivastava [7]. Motivated by earlier works of Aouf et al.,[] and Dziok and Raina[4] we define the following new subclass of T involving hypergeometric functions. For λ, < β, B < A, γ,we let HF λ γ(α, β, A, B) denote the subclass of T consisting of functions of the form (.2) satisfying the analytic condition where (B A)γ zf λ (z) F λ (z) [ zf λ (z) F λ (z) α ] B [ zf λ (z) ] < β, z U (.7) F λ (z) zf λ (z) F λ (z) = zhf (z) + λz 2 Hf (z) ( λ)h + λzhf (z), λ (.8)

3 Neighborhood and partial sums 73 and H = z + a n Γ n z n (.9) where Γ n is given by (.6) The main object of the present paper is to investigate (n, δ)- neighborhoods of functions HF λ γ(α, β, A, B). Furthermore, we obtain Partial sums f k (z) and Integral means inequality of functions in the class HF λ γ(α, β, A, B). We state the following Lemma, due to Vijaya and Deppa [5] which provide the necessary and sufficient conditions for functions HF λ γ(α, β, A, B). Lemma.. A function T is in the class HF λ γ (α, β, A, B) if and only if where (n, λ, α, β, γ)a k (.) (n, λ, α, β, γ) = ( + nλ λ)[(n )( βb) + βγ(b A)(n α)]γ n. (.) βγ(b A)( α) 2. Neighborhood results In this section we discuss neighborhood results of the class HF λ γ(α, β, A, B) due to Goodman [5] and Ruscheweyh []. We define the δ neighborhood of function T. Definition 2.. For f T of the form (.2) and δ. We define a δ neighbourhood of a function by { } N δ (f) = g : g T : g(z) = z c n z n and n a n c n δ. (2.) In particular, for the identity function e(z) = z, we immediately have { } N δ (e) = g : g T : g(z) = z c n z n and n c n δ. (2.2) Theorem 2.2. If δ = 2 (2,λ,α,β,γ) then HF λ γ (α, β, A, B) N δ (e), where (2, λ, α, β, γ) = ( + λ)[( βb) + βγ(b A)(2 α)]γ 2. (2.3) βγ(b A)( α) Proof. For a function HFγ λ (α, β, A, B) of the form (.2), Lemma. immediately yields, ( + nλ λ)[(n )( βb) + βγ(b A)(n α)]γ n a n βγ(b A)( α)

4 74 K. Vijaya and K. Deepa ( + λ)[ βb + βγ(b A)(2 α)]γ 2 a n βγ(b A)( α) βγ(b A)( α) a n = ( + λ)[ βb + βγ(b A)(2 α)]γ 2 (2.4) (2, λ, α, β, γ). On the other hand, we find from (.) and (2.4) that ( + nλ λ)[(n )( βb) + βγ(b A)(n α)]γ n a n βγ(b A)( α). That is Hence βγ(b A)( α)[ βb( + λ λ + )] na n ( + λ)[ βb + βγ(b A)(2 α)]( βb). na n 2 = δ. (2, λ, α, β, γ) A function f T is said to be in the class HF λ γ(ρ, α, β, A, B) if there exists a function h HF λ γ(ρ, α, β, A, B) such that h(z) < ρ, (z U, ρ < ). (2.5) Now we determine the neighborhood for the class HF λ γ(ρ, α, β, A, B). Theorem 2.3. If h HF λ γ(ρ, α, β, A, B) and ρ = δφa B (2, λ, α, β, γ) 2[ (2, λ, α, β, γ) ] (2.6) then N δ (h) HF λ γ(ρ, α, β, A, B) where (2, λ, α, β, γ) is defined in (2.3). Proof. Suppose that f N δ (h) we then find from (2.) that n a n d n δ which implies that the coefficient inequality a n d n δ 2. Next, since h HF λ γ(α, β, A, B), we have d n = (2, λ, α, β, γ)

5 Neighborhood and partial sums 75 so that h(z) < a n d n d n δ 2 φa B (2, λ, αβ, γ) (2, λ, α, β, γ) δ (2, λ, αβ, γ) 2( (2, λ, αβ, γ) ) = ρ provided that ρ is given precisely by (2.6). Thus by definition, f HF λ γ(ρ, α, β, A, B) for ρ given by (2.6), which completes the proof. 3. Partial sums Silverman [4] determined the sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions }, viz., R {/f k (z)}, R {f k (z)/}, R {f (z)/f k {f (z)} and Re k (z)/f (z) for their sequences of partial sums f k (z) = z + k a n z n of the analytic function = z + a n z n. In the following theorems we discuss results on partial sums for functions HF λ γ (α, β, A, B). Theorem 3.. If f of the form (.2) satisfies the condition (.), then R { } f k (z) φa B (k + 2, λ, α, β, γ) (k + 2, λ, α, β, γ) (3.) and R { } fk (z) φa B (k + 2, λ, α, β, γ) (k + 2, λ, α, β, γ) + (3.2) where (k + 2, λ, α, β, γ) is given by (.). The results are sharp for every k, with the extremal function given by = z (k + 2, λ, α, β, γ)zn+. (3.3) Proof. In order to prove (.), it is sufficient to show that [ ] (k + 2, λ, α, β, γ) f k (z) φa B (k + 2, λ, α, β, γ) (k + 2, λ, α, β, γ) + z z (z U)

6 76 K. Vijaya and K. Deepa we can write a n z n (k + 2, λ, α, β, γ) a n z n = (k + 2, λ, α, β, γ) a n z n a n z n k (k + 2, λ, α, β, γ) (k + 2, λ, α, β, γ) n=k+ a φa B nz n (k + 2, λ, α, β, γ) (k + 2, λ, α, β, γ) Then w(z) = Obviously w() = and w(z) if and only if = + w(z) w(z). (k + 2, λ, α, β, γ) n=k+ a nz n 2 2 k a nz n (k + 2, λ, α, β, γ) n=k+ a nz. n 2(k + 2, λ, α, β, γ) (k+2,λ,α,β,γ) n=k+ an 2 2 k an φa B (k+2,λ,α,β,γ) n=k+ which is equivalent to a n + (k + 2, λ, α, β, γ) a n 2 2 n=k+ n=k+ an. Now, w(z) a n a n. (3.4) In view of (.), this is equivalent to showing that ((n, λ, α, β, γ) )a n + ((n, λ, α, β, γ) (k + 2, λ, α, β, γ))a n. n=k+ To see that the function f given by (3.3) gives the sharp results, we observe for z = re 2πi n that f k (z) = (k + 2, λ, α, β, γ)zn (k + 2, λ, α, β, γ) where r. Thus, we have completed the proof of (3.). The proof of (3.2) is similar to (3.) and will be omitted. Theorem 3.2. If of the from (.2) satisfies (.) then { } f (z) R f k (z) φa B (k + 2, λ, α, β, γ) k (k + 2, λ, α, β, γ) (3.5)

7 Neighborhood and partial sums 77 and R { } f k (z) (k + 2, λ, α, β, γ) f (z) (k + 2, λ, α, β, γ) + k + (3.6) where (k + 2, λ, α, β, γ) is given by (.). The results are sharp for every k, with the extremal function given by (3.3). Proof. In order to prove (3.5) it is sufficient to show that [ ] (k + 2, λ, α, β, γ) f (z) k + f k (z) φa B (k + 2, λ, α, β, γ) n (k + 2, λ, α, β, γ) we can write Then w(z) = (k + 2, λ, α, β, γ) k + = φa B (k + 2, λ, α, β, γ) k + = + w(z) w(z). [ f (z) f k (z) φa B [ na nz n k na nz ] (k + 2, λ, α, β, γ) n (k + 2, λ, α, β, γ) φab n + z z (z U) (k + 2, λ, α, β, γ) k (k + 2, λ, α, β, γ) (k + 2, λ, α, β, γ)(k + ) n=k+ na nz n 2 2 k na nz n (k + 2, λ, α, β, γ)(k + ) n=k+ na nz. n Obviously w() = and w(z) Now, w(z) if and only if which is equivalent to (k + 2, λ, α, β, γ)(n + ) n=k+ na n 2 2 k na n (k + 2, λ, α, β, γ)(n + ) n=k+ na. n 2 φa B (k + 2, λ, α, β, γ) (k + ) n=k+ na n + φa B (k + 2, λ, α, β, γ) (k + ) In view of (.), this is equivalent to showing that ((n, λ, α, β, γ) n)a n + n=k+ na n 2 2 n=k+ na n na n. (3.7) ( ) (n, λ, α, β, γ) φa B (k + 2, λ, α, β, γ) a n (k + )n which completes the proof of (3.5). The proof of (3.6) is similar to (3.5) and hence we omit proof. ]

8 78 K. Vijaya and K. Deepa 4. Integral means inequality In 975, Silverman [3] 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 Silverman [2] that 2π f(re iθ ) 2π η dθ f 2 (re iθ ) η dθ, for all f T, η > and < r <. and settled in Silverman (997), also proved his conjecture for the subclasses S (α) and K(α) of T. Lemma 4.. If and g(z) are analytic in U with g(z), then 2π f(re iθ ) µ dθ where µ, z = re iθ and < r <. 2π g(re iθ ) µ dθ Application of Lemma 4. to function of in the class HF λ γ (α, β, A, B) gives the following result. Theorem 4.2. Let µ >. If HFγ λ (α, β, A, B) is given by (.2) and f 2 (z) is defined by f 2 (z) = z (2, λ, α, β, (4.) γ)z2 where (2, λ, α, β, γ) is defined by (2.3).Then for z = reiθ, < r <, we have 2π 2π η dθ f 2 (z) η dθ. (4.2) Proof. For functions f of the form (.2) is equivalent to proving that 2π η 2π a n z n dθ (2, λ, α, β, γ)z By Lemma 4., it suffices to show that Setting a n z n a n z n = (2, λ, α, β, γ)z. η dθ. (4.3) (2, λ, α, β, γ)w(z),

9 Neighborhood and partial sums 79 and using (.), we obtain w(z) = φ A B(n, λ, α, β, γ)a n z n z (n, λ, α, β, γ)a n z, where (n, λ, α, β, γ) is given by (. ) which completes the proof. Remark 4.3. We observe that for λ =, if µ = the various results presented in this chapter 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 chapter are fairly straight- forward. Acknowledgements. The authors record their sincere thanks to Prof. G. Murugusundaramoorthy, VIT University, Vellore-6324, India and anonymous referees for their valuable suggestions. References [] Aouf, M.K. and Murugusundaramoorthy, G., On a subclass of uniformly convex functions defined by the Dziok-Srivastava Operator, Austral. J. Math. Anal. and Appl., 3(27), -2. [2] Carlson, B.C. and Shaffer, D.B., Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 5(984), [3] Dziok, J. and Srivastava, H.M., Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform and Spec. Funct., 4(23), 7-8. [4] Dziok, J. and Raina, R.K., Families of analytic functions associated with the Wright generalized hypergeometric function, Demonstratio Math., 37(24), no. 3, [5] Goodman, A.W., Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8(957), [6] Murugusundaramoorthy, G. and Magesh, N., Starlike and convex functions of complex order involving the Dziok-Srivastava operator, Integral Transforms and Special Functions, 8(27), [7] Owa, S. and Srivastava, H.M., Univalent and Starlike generalized hypergeometric functions, Canad. J. Math., 39(987), [8] Ri-Guang Xiang, Zhi-Gang Wang, Shao-Mou Yuan, A class of multivalent analytic functions involving the Dziok-Srivastava operator, Stud. Univ. Babes-Bolyai, Math., 55(2), no. 2, [9] Ruscheweyh, S., New criteria for univalent functions, Proc. Amer. Math. Soc., 49(975), 9 5. [] Ruscheweyh, S., Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 8 (98),

10 8 K. Vijaya and K. Deepa [] Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 5(975), 9-6. [2] Silverman, H., A survey with open problems on univalent functions whose coefficients are negative, Rocky Mountain J. Math., 2(99), [3] Silverman, H., Integral means for univalent functions with negative coefficients, Houston J. Math., 23(997), [4] Silverman, H., Partial sums of starlike and convex functions, J. Math. Anal. and Appl., 29(997), [5] Vijaya, K. and Deepa, K., Subclass of univalent functions with negative coefficients involving Dziok-Srivastava Operator, (Communicated). K. Vijaya School of Advanced Sciences VIT University Vellore, 6324, Tamilnadu, India kvijaya@vit.ac.in K. Deepa School of Advanced Sciences VIT University Vellore, 6324, Tamilnadu, India