On the Study of Starlike Functions Associated with the Generalized Sine Hyperbolic Function

Baseer Gul, Muhammad Arif, Reem K. Alhefthi, Daniel Breaz, Luminiţa-Ioana Cotîrlă () and Eleonora Rapeanu
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Baseer Gul: Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
Muhammad Arif: Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
Reem K. Alhefthi: Department of Mathematics, College of Sciences, King Saudi University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Daniel Breaz: Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania
Luminiţa-Ioana Cotîrlă: Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Eleonora Rapeanu: Department of Mathematics, “Mircea cel Batran” Naval Academy, 900218 Constanţa, Romania

Mathematics, 2023, vol. 11, issue 23, 1-22

Abstract: Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since ℜ ( 1 + sinh ( z ) ) ≯ 0 , it implies that the class S sinh * introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter λ with the restriction 0 ≤ λ ≤ ln ( 1 + 2 ) , and by doing that, ℜ ( 1 + sinh ( λ z ) ) > 0 . The present research intends to provide a novel subclass of starlike functions in the open unit disk U , denoted as S sinh λ * , and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients a n for n = 2 , 3 , 4 , 5 . Then, we prove a lemma, in which the largest disk contained in the image domain of q 0 ( z ) = 1 + sinh ( λ z ) and the smallest disk containing q 0 ( U ) are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including S * ( β ) and K ( β ) of starlike functions of order β and convex functions of order β . Investigating S sinh λ * radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding S sinh λ * radii of different subclasses is the calculation of that value of the radius r < 1 for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of λ , is also obtained.

Keywords: starlike functions; Janowski starlike function; sine hyperbolic function; radii problems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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