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30 Subordination Results for Certain Subclasses of Analytic Functions and 2(β − 1) + 1 − λ + 3 − 2β − λ(2β − 1) Γ 2(α 1 ) Re{f(z)} > − [1 − λ + |3 − 2β − λ(2β − 1)|]Γ 2 (α 1 ) . The constant factor [1−λ+|3−2β−λ(2β−1)|]Γ 2 (α 1 ) 2{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]Γ 2 (α 1 )} in the (2.25) subordination result (2.24) can not be replaced by a larger one and the function 2(β − 1) f 0 (z) = z − [1 − λ + |3 − 2β − λ(2β − 1)|]Γ 2 (α 1 ) z2 gives the sharpness. (2.26) By taking b k = l+1+μ(k−1) m (m ∈ N l+1 0 , μ, l ≥ 0) in Lemma 2 and Theorem 1, we obtain the following corollary: Corollary 6. Let the function f(z) defined by (1.1) be in the class T ∗ (m, μ, l, λ, β) and satisfy the condition ∞ (1 − λ)(k − 1) l + 1 + μ(k − 1) k − (2β − 1) + [1 + λ(k − 1)] l + 1 k=2 Then for every function h ∈ K, we have: m |a k | ≤ 2(β − 1). (2.27) [1 − λ + |3 − 2β − λ(2β − 1)|](l + 1 + μ) m (f ∗ h)(z) ≺ h(z) 2(l + 1) m (β − 1) 2 + 1 − λ + 3 − 2β − λ(2β − 1) (l + 1 + μ)m and (2.28) 2(l + 1) m (β − 1) 3 − 2β + 1 − λ + (l + 1 + μ)m −λ(2β − 1) Re{f(z)} > − [1 − λ + |3 − 2β − λ(2β − 1)|](l + 1 + μ) m. (2.29) The constant factor [1−λ+|3−2β−λ(2β−1)|](l+1+μ) m 2{2(l+1) m (β−1)+[1−λ+|3−2β−λ(2β−1)|](l+1+μ) m } in the subordination result (2.28) can not be replaced by a larger one and the function 2(β − 1)(l + 1) m f 0 (z) = z − [1 − λ + |3 − 2β − λ(2β − 1)|](l + 1 + μ) m z2 (2.30) gives the sharpness. By taking b k = C k (b, μ), where C k (b, μ) defined by (1.10), in Lemma 2 and Theorem 1, we obtain the following corollary: Corollary 7. Let the function f(z) defined by (1.1) be in the class T ∗ (μ, b, λ, β) and satisfy the condition ∞ (1 − λ)(k − 1) k − (2β − 1) + [1 + λ(k − 1)] C k (b, μ)|a k | ≤ 2(β − 1). k=2 Then for every function h ∈ K, we have: (2.31) [1 − λ + |3 − 2β − λ(2β − 1)|]C 2 (b, μ) (f ∗ h)(z) ≺ h(z) 2(β − 1) 2 3 − 2β + 1 − λ + −λ(2β − 1) C 2 (b, μ) and (2.32) 2(β − 1) 3 − 2β + 1 − λ + −λ(2β − 1) C 2(b, μ) Re{f(z)} > − 1 − λ + 3 − 2β − . λ(2β − 1) C 2(b, μ) The constant factor [1−λ+|3−2β−λ(2β−1)|]C 2 (b,μ) 2{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]C 2 (b,μ)} in the (2.33) subordination result (2.32) can not be replaced by a larger one and the function 2(β − 1) f 0 (z) = z − [1 − λ + |3 − 2β − λ(2β − 1)|]C 2 (b, μ) z2 gives the sharpness. 3. ACKNOWLEDGEMENTS (2.34) The authors thank the anonymous referees of the paper for their helpful suggestions. 4. REFERENCES 1. Aouf, M.K., A.A.. Shamandy, A.O. Mostafa & E.A. Adwan. Subordination results for certain class of analytic functions defined by convolution.
M.K. Aouf et al 31 Rend. del Circolo Math. di Palermo (in press). 2. Bulboaca, T. Differential Subordinations and Superordinations, Recent Results. House of Scientific Book Publ., Cluj-Napoca (2005). 3. A Cătaş, G.I. Oros & G. Oros. Differential subordinations associated with multiplier transformations. Abstract Appl. Anal. ID845724: 1-11 (2008). 4. Dziok, J. & H.M. Srivastava. Classes of analytic functions with the generalized hypergeometric function. Appl. Math. Comput. 103: 1-13 (1999). 5. Dziok, J. & , H.M. Srivastava. Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transform. Spec. Funct. 14: 7-18 (2003). 6. Miller, S.S. & P.T. Mocanu. Differential Subordinations Theory and Applications. In: Series on Monographs and Textbooks in Pure and Applied Mathematics 255. Marcel Dekker, New York (2000). 7. Nishiwaki, J. & S Owa.. Coefficient inequalities for certain analytic functions, Internat. J. Math. Math. Sci. 29 (5): 285-290 (2002). 8. Owa, S. & J. Nishiwaki. Coeffocient estimates for certain classes of analytic functions. J. Inequal. Pure Appl. Math. 3 (5), Art. 72: 1-12 (2002). 9. Owa, S. & H.M. Srivastava. Some generalized convolution properties associated with certain subclasses of analytic functions. J. Inequal. Pure Appl. Math. 3 (3), Art. 42: 1-27 (2002). 10. Srivastava, H.M. & A.A Attiya. Some subordination results associated with certain subclasses of analytic functions. J. Inequal. Pure Appl. Math. 5 (4), Art. 82: 1-14 (2004). 11. Srivastava, H.M. & A.A Attiya.. An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination. Integral Transform. Spec. Funct. 18: 207-216 (2007). 12. Srivastava, H.M. & S. Owa. Current Topics in Analytic Function Theory. World Scientific Publishing Company, Singapore (1992). 13. Uralegaddi, BA. & A.R. Desa. Convolutions of univalent functions with positive coefficients. Tamkang J. Math. 29: 279-285 (1998). 14. Uralegaddi, B.A., M.D. Ganigi & S.M Sarangi. Univalent functions with positive coefficients. Tamkang J. Math. 25: 225-230 (1994). 15. Wilf, S. Subordinating factor sequence for convex maps of the unit circle. Proc. Amer. Math. Soc. 12: 689-693 (1961).
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