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New Subclass of Convex Functions Concerning Infinite Cone

Document Type : Research Paper

Authors

1 Department of Mathematics, Payame Noor University, P.O.Box 19395-3697 Tehran, Iran.

2 Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.

Abstract

We introduce a new subclass of convex functions as follows:
\[
  \mathcal{K}_{IC}:=\left\{f\in \mathcal{A}:{\rm
  Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>\left|f'(z)-1\right|,\quad
  |z|<1\right\},
\]
where $\mathcal{A}$ denotes the class of analytic and normalized functions in the unit disk $|z|<1$. Some properties of this particular class, including subordination relation, integral representation, the radius of convexity, rotation theorem, sharp coefficients estimate and Fekete-Szeg\"{o} inequality associated with the $k$-th root transform, are investigated.

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References
1. R.M. Ali, S.K. Lee, V. Ravichandran and S. Supramanian, The Fekete-Szegö coefficient functional for transforms of analytic functions, Bull. Iranian Math. Soc., 35 (2009), pp. 119-142.
2. L. Bieberbach, Aufstellung und Beweis des Drehungssatzes fürschlichte konforme Abbildungen, Math. Zeit., 4 (1919), pp. 295-305.
3. M. Fekete and G. Szegö, Eine Bemerkung über ungerade schlichte Funktionen, Polon. Math., 1 (2) (1933), pp. 85-89.
4. A.W. Goodman, On uniformly convex functions, Ann. Polon. Math., 56 (1991), pp. 87-92.
5. A.S. Issa and M. Darus,On the Fekete-Szegö problem associated with generalized fractional operator, Int. J. Nonlinear Anal. Appl., 14 (1) (2023), pp. 25-32.
6. R. Kargar, A. Ebadian and J. Sokół, On Booth lemniscate and starlike functions, Anal. Math. Phys., 9 (2019), pp. 143–154.
7. F.R. Keogh and E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), pp. 8-12.
8. W. Ma and D. Minda, Uniformly convex functions, Ann. Pol. Math., 2 (57) (1992), pp. 165-175.
9. W. Ma and D. Minda, Uniformly convex functions II, Ann. Pol. Math., 3 (58) (1993), pp. 275-285.
10. S.N. Malik, S. Riaz, M. Raza and S. Zainab, Fekete-Szegö Problem of Functions Associated with Hyperbolic Domains, Sahand Commun. Math. Anal., 14 (1) (2019), pp. 73-88.
11. S.S. Miller and P.T. Mocanu, Differential Subordinations, Theory and Applications, Series of Monographs and Textbooks in Pure and Applied Mathematics, 225, Marcel Dekker Inc., New York / Basel 2000.
12. K. Noshiro, On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ., 2 (1934-35), pp. 129-155.
13. M. Nunokawa, On properties of non-Carathéodory functions, Proc. Jpn. Acad., Ser. A, Math. Sci., 68 (1992), pp. 152-153.
14. M.I.S. Robertson, On the theory of univalent functions, Ann. Math., 37 (1936), pp. 374-408.
15. F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), pp. 189-196.
16. St. Ruscheweyh, Extension of Szegö’s theorem on the sections of univalent functions, SIAM J. Math. Anal., 19 (1988), pp. 1442-1449.
17. St. Ruscheweyh and V. Singh, On a Briot-Bouquet equation related to univalent functions, Rev. Roumaine Math. Pures Appl., 24 (1979), pp. 285-290.
18. J. Sokół and M. Nunokawa, On some class of convex functions, C. R. Acad. Sci. Paris, Ser., I 353 (2015), pp. 427-431.
19. H.M. Srivastava, D. Răducanu and P. Zaprawa, A certain subclass of analytic functions defined by means of differentiona subordination, Filomat, 30:14 (2016), pp. 3743-3757.
20. S. Warschawski, On the higher derivatives at the boundary in conformal mappings, Trans. Amer. Math. Soc., 38 (1935), pp. 310-340.
Sahand Communications in Mathematical Analysis
Volume 21, Issue 2
March 2024
Pages 167-178
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