Generalization of Quasi Convex Functions using Convolution
Abstract
In this paper, an up-to-date generalization of the class $\mathtt{C^\star}$ of quasi-convex functions is given by introducing new class $\mathfrak{\mathtt{C}^\star_g[a, b]}$. Furthermore its basic properties, its relationship with other subclasses of $\mathtt{S}$, inclusion relations and some other interesting properties are derived.
References
O. Altintaş and Ö. Ö. Kiliç, Coefficient estimates for a class containing quasi-convex functions, Turk. J. Math. 42 (2018), 2819-2825. https://doi.org/10.3906/mat-1805-90
S. M. Barnardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429-446. https://doi.org/10.1090/S0002-9947-1969-0232920-2
P. L. Duren, Univalent Functions, Springer-Verlag, Berlin, 1983.
A. W. Goodman, Univalent Functions, Vols. I and II, Polygonal Publishing House, Washington, NJ, 1983.
W. K. Hayman, Multivalent Functions, Cambridge University Press, U.K., 1967.
W. Janowski, Some extremal problems for certain families of analytic functions I, Ann. Polon. Math. 28 (1973), 297-326. https://doi.org/10.4064/ap-28-3-297-326
R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755-758. https://doi.org/10.1090/S0002-9939-1965-0178131-2
S. S. Miller, Differential Inequalities and Carathéodory functions, Bull. Amer. Math. Soc. 81 (1975), 79-81. https://doi.org/10.1090/S0002-9904-1975-13643-3
S. S. Miller and P. T. Mocanu, Differential Subordination: Theory and Applications, Vol. 225, Marcel Dekker Inc., New York, Basel, 2000. https://doi.org/10.1201/9781482289817
K. I. Noor and D. K. Thomas, On quasi convex univalent functions, Inter. J. Math. Math. Sci. 3 (1980), 255-266. https://doi.org/10.1155/S016117128000018X
K. I. Noor, On quasi convex functions and related topics I, J. Math. Math. Sci. 10(2) (1987), 241-258. https://doi.org/10.1155/S0161171287000310
Ch. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Gottingen, 1975.
M. S. Robertson, On the theory of univalent functions, Ann. of Math. 28 (1936), 297-326.
St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135. https://doi.org/10.1007/BF02566116
S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109-115. https://doi.org/10.1090/S0002-9939-1975-0367176-1
K. Sakaguchi and S. Fukui, An extension of a theorem of S. Ruscheweyh, Bull. Fac. Edu. Wakayama Univ. Nat. Sci. 29 (1980), 1-3.
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