Abstract
Making use of the convolution product, we introduce a unified class of analytic functions with negative coefficients. Also, we obtain the coefficient bounds, extreme points, and radius of starlikeness for functions belonging to the generalized class P g T (λ, α, β). Furthermore, partial sums f k (z) of functions f(z) in the class P g (λ, α, β) are considered and sharp lower bounds for the ratios of the real parts of f(z) to f k (z) and f′(z) to f′ k (z) are determined. Relevant connections of the results with various known results are also considered.
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References
O. P. Ahuja, Integral Operators of Certain Univalent Functions, Internat. J. Math. Soc. 8, 653 (1985).
O. P. Ahuja, On the Generalized Ruscheweyh Class of Analytic Functions of Complex Order, Bull. Austral. Math. Soc. 47, 247 (1993).
R. Bharati, R. Parvatham, and A. Swaminathan, On Subclasses of Uniformly Convex Functions and Corresponding Class of Starlike Functions, Tamkang J. Math. 26(1), 17 (1997).
B. C. Carlson and D. B. Shaffer, Starlike and Prestarlike Hypergeometric Functions, SIAM J. Math. Anal. 15, 737 (1984).
N. E. Cho and T. H. Kim, Multiplier Transformations and Strongly Close-to-convex Functions, Bull. Korean Math. Soc. 40(3), 399 (2003).
N. E. Cho and H. M. Srivastava, Argument Estimates of Certain Analytic Functions Defined by a Class of Multiplier Transformations, Math. Comput. Modelling 37(1–2), 39 (2003).
A. W. Goodman, On Uniformly Convex Functions, Ann. Polon. Math. 56, 87 (1991).
A. W. Goodman, On Uniformly Starlike Functions, J. Math. Anal. Appl. 155, 364 (1991).
G. Murugusundaramoorthy and N. Magesh, Linear Operators Associated with a Subclass of Uniformly Convex Functions, Inter. J. Pure and Appl. Math. 3(1), 123 (2006).
F. Rønning, Uniformly Convex Functions and a Corresponding Class of Starlike Functions, Proc. Amer. Math. Soc. 118, 189 (1993).
F. Rønning, Integral Representations for Bounded Starlike Functions, Annal. Polon. Math. 60, 289 (1995).
T. Rosy, K. G. Subramanian, and G. Murugusundaramoorthy, Neighbourhoods and Partial sums of Starlike Functions Based on Ruscheweyh Derivatives, J. Ineq. Pure and Appl. Math. 4(64), 4 (2003).
T. Rosy and G. Murugusundaramoorthy, Fractional Calculus and Their Applications to Certain Subclass of Uniformly Convex Functions, Far East. J. Math. Sci. 15(2), 231 (2004).
S. Ruscheweyh, New Criteria for Univalent Functions, Proc. Amer. Math. Soc. 49, 109–115 (1975).
S. Shams and S. R. Kulkarni, On a Class of Univalent Functions Defined by Ruscheweyh Derivatives, Kyungpook Math. J. 43, 579 (2003).
H. Silverman, Univalent Functions with Negative Coefficients, Proc. Amer.Math. Soc. 51, 109 (1975).
H. Silverman, Partial Sums of Starlike and Convex Functions, J. Math. Anal. Appl. 209, 221 (1997).
E. M. Silvia, Partial Sums of Convex Functions of order α, Houston. J. Math., Math. Soc. 11(3), 397 (1985).
G. Ş. Sălăgean, Subclasses of Univalent Functions, in Complex analysis—fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), p. 362, Lecture Notes in Math., 1013, Springer, Berlin.
K. G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahmanyam, and H. Silverman, Subclasses of Uniformly Convex and Uniformly Starlike Functions, Math. Japonica 42(3), 517 (1995).
K. G. Subramanian, T. V. Sudharsan, P. Balasubrahmanyam, and H. Silverman, Classes of Uniformly Starlike Functions, Publ. Math. Debrecen. 53(3–4), 309 (1998).
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Murugusundaramoorthy, G., Rosy, T. & Muthunagai, K. A unified class of analytic functions with negative coefficients. Lobachevskii J Math 29, 175–185 (2008). https://doi.org/10.1134/S1995080208030086
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DOI: https://doi.org/10.1134/S1995080208030086