Harmonic functions with missing coefficients

Year 2019, Volume: 48 Issue: 6, 1778 - 1791, 08.12.2019

Jacek Dziok

https://doi.org/10.15672/HJMS.2018.637

Cited By: 1

Abstract

In the paper we introduce the classes of functions with missing coefficients defined by generalized Ruscheweyh derivatives and we show that they can be presented as dual sets. Moreover, by using extreme points theory, we obtain estimations of classical convex functionals on the defined classes of functions. Some applications of the main results are also considered.

Keywords

Harmonic functions, Subordination, Missing coefficients, Ruscheweyh operator, Dual sets, Correlated coefficients

References

• [1] O. Ahuja, Connections between various subclasses of planar harmonic mappings involving hypergeometric functions, Appl. Math. Comput. 198, 305–316, 2008.
• [2] O.P. Ahuja and J.M. Jahangiri, Certain multipliers of univalent harmonic functions, Appl. Math. Lett. 18, 1319–1324, 2005.
• [3] M. Darus and K. Al-Shaqsi, K. On certain subclass of harmonic univalent functions, J. Anal. Appl. 6, 17–28, 2008.
• [4] J. Dziok, On Janowski harmonic functions, J. Appl. Anal. 21, 99–107, 2015.
• [5] J. Dziok, Ruscheweyh-type harmonic functions with correlated coefficients, Filomat, 33 (12), 2019.
• [6] J. Dziok, J.M. Jahangiri, and H. Silverman, Harmonic functions with varying coefficients, J. Inequal. Appl. 139, 2016.
• [7] J. Dziok, M. Darus, J. Sokół, and T. Bulboaca, Generalizations of starlike harmonic functions, C. R. Math. Acad. Sci. Paris, 354, 13–18, 2016.
• [8] D.J. Hallenbeck and T.H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman Advanced Publishing Program, Boston, Pitman, 1984.
• [9] J.M. Jahangiri, Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235, 470–477, 1999.
• [10] J.M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, Starlikeness of harmonic functions defined by Ruscheweyh derivatives, J. Indian Acad. Math. 26, 191–200, 2004.
• [11] J.M. Jahangiri and H. Silverman, Harmonic univalent functions with varying arguments, Int. J. Appl. Math. 8, 267–275, 2002.
• [12] W. Janowski, Some extremal problems for certain families of analytic functions I, Ann. Polon. Math. 28, 297–326, 1973.
• [13] S.Y. Karpuzogullari, M. Öztürk, and M. Yamankaradeniz, A subclass of harmonic univalent functions with negative coefficients, Appl. Math. Comput. 142 (2-3), 469– 476, 2003.
• [14] M. Krein and D. Milman, On the extreme points of regularly convex sets, Studia Mathematica 9, 133–138, 1940.
• [15] H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42, 689–692, 1936.
• [16] P. Montel, Sur les families de functions analytiques qui admettent des valeurs exceptionelles dans un domaine, Ann. Sci. Ecole Norm. Sup. 23, 487–535, 1912.
• [17] G. Murugusundaramoorthy and K. Vijaya, On certain classes of harmonic functions involving Ruscheweyh derivatives, Bull. Calcutta Math. Soc. 96, 99–108, 2004.
• [18] M. Öztürk, S. Yalçin, and M. Yamankaradeniz, Convex subclass of harmonic starlike functions, Appl. Math. Comput. 154, 449–459, 2004.
• [19] S. Ruscheweyh, Convolutions in geometric function theory, in: Sem. Math. Sup. 83, Les Presses del’Université de Montréal, 1982.
• [20] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math Soc. 49, 109–115, 1975.
• [21] H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl. 220, 283–289, 1998.
• [22] S. Yalçın, S.B. Joshi, and E. Yaşar, On certain subclass of harmonic univalent functions defined by a generalized Ruscheweyh derivatives operator, Appl. Math. Sci. (Ruse) 4, 327–336, 2010.
• [23] S. Yalçın and M. Öztürk, Harmonic functions starlike of the complex order, Mat. Vesnik 58, 7–11, 2006.
• [24] S. Yalçın, M. Öztürk, and M. Yamankaradeniz, On the subclass of Salagean-type harmonic univalent functions, J. Inequal. Pure Appl. Math. 8, Art. 54, 9 pp., 2007.