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Convolution properties of a class of starlike functions


Authors: Ram Singh and Sukhjit Singh
Journal: Proc. Amer. Math. Soc. 106 (1989), 145-152
MSC: Primary 30C45
DOI: https://doi.org/10.1090/S0002-9939-1989-0994388-6
MathSciNet review: 994388
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Abstract: Let $ R$ denote the class of functions $ f(z) = z + {a_2}{z^2} + \cdots $ that are analytic in the unit disc $ E = \{ z:\left\vert z \right\vert < 1\} $ and satisfy the condition $ \operatorname{Re} (f'(z) + zf''(z)) > 0,z \in E$. It is known that $ R$ is a subclass of $ {S_t}$, the class of univalent starlike functions in $ E$. In the present paper, among other things, we prove (i) for every $ n \geq 1$, the $ n$th partial sum of $ f \in R,{s_n}(z,f)$, is univalent in $ E$, (ii) $ R$ is closed with respect to Hadamard convolution, and (iii) the Hadamard convolution of any two members of $ R$ is a convex function in $ E$.


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  • [1] P. N. Chichra, New subclasses of the class of close-to-convex functions, Proc. Amer. Math. Soc. (1) 62 (1977), 37-43. MR 0425097 (54:13055)
  • [2] L. Fejěr, Uber die positivitat von summen, die nach trigonometrischen order Legendreschen funktionen fortschreiten, Acta Litt. Ac Sci. Szeged (1925), 75-86.
  • [3] I. S. Jack, Functions starlike and convex of order $ \alpha $, J. London Math Soc. (2) 3 (1971), 469-474. MR 0281897 (43:7611)
  • [4] J. Krzyź, A counter example concerning univalent functions, Mat. Fiz. Chem. 2 (1962), 57-58.
  • [5] M. S. Robertson, An extremal problem for functions with positive real part, Michigan Math. J. 11 (1964), 327-335. MR 0170002 (30:243)
  • [6] W. Rogosinski and G. Szegö, Uber die Abschimlte Von potenzreihen die in ernein Kreise be schrankt bleiben, Math. Z., 28 (1928), 73-94. MR 1544940
  • [7] St. Ruscheweyh and T. Sheil Small, Hadamard products of schlicht functions and the Polya-Schoenberg conjecture, Comm. Math. Helv. 48 (1973), 119-135. MR 0328051 (48:6393)
  • [8] R. Singh and S. Singh, Starlikeness and convexity of certain integrals, Ann. Univ. Mariae Curie-Skłodowska Sect. A 35 (1981), 45-47. MR 763928 (86a:30024)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0994388-6
Keywords: Univalent, starlike and convex functions, convex null sequences
Article copyright: © Copyright 1989 American Mathematical Society

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