In the paper we introduce an idea of harmonic functions with correlated coefficients which generalize the ideas of harmonic functions with negative coefficients introduced by Silverman and harmonic functions with varying coefficients defined by Jahangiri and Silverman. Next we define classes of harmonic functions with correlated coefficients in terms of generalized Dziok-Srivastava operator. 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.
Jacek DZIOK
. HARMONIC FUNCTION WITH CORRELATED COEFFICIENTS[J]. Acta mathematica scientia, Series B, 2019
, 39(6)
: 1661
-1673
.
DOI: 10.1007/s10473-019-0615-6
[1] Ahuja O. Connections between various subclasses of planar harmonic mappings involving hypergeometric functions. Appl Math Comput, 2008, 198:305-316
[2] Ahuja O P, Jahangiri J M. Certain multipliers of univalent harmonic functions. Appl Math Lett, 2005, 18:1319-1324
[3] Al-Hawary T, Frasin B A, Darus M. Fekete-Szegö problem for certain classes of analytic functions of complex order defined by the Dziok-Srivastava operator. Acta Math Vietnam, 2014, 39:185-192
[4] Al-Khal R A. Goodman-Rřnning-type harmonic univalent functions based on Dziok-Srivastava operator. Appl Math Sci, 2011, 5:573-584
[5] Al-Kharsani H A, Al-Khal R A. Univalent harmonic functions. J Inequal Pure Appl Math, 2007, 8(2):Article 59, 8 pp
[6] Al-Shaqsi K, Darus M, Fadipe-Joseph O A. A new subclass of Salagean-type harmonic univalent functions. Abstr Appl Anal, 2010, Art ID 821531, 12 pp
[7] Aouf M K, Srivastava H M. Some families of starlike functions with negative coefficients. J Math Anal Appl, 1996, 203:762-790
[8] Darus M, Al-Shaqsi K. On certain subclass of harmonic univalent functions. J Anal Appl, 2008, 6:17-28
[9] Dixit K K, Porwal S. Convolution of the subclass of Salagean-type harmonic univalent functions with negative coefficients. Gen Math, 2010, 18:59-6
[10] Dziok J. Classes of harmonic functions associated with Ruscheweyh derivatives. Rev R Acad Cienc Exactas Fís Nat Ser A Math, 2018(DOI:10.1007/s13398-018-0542-8)
[11] Dziok J, Jahangiri J M, Silverman H. Harmonic functions with varying coefficients. J Inequal Appl, 2016, 2016:139
[12] Dziok J, Darus M, Sokół J, Bulboaca T. Generalizations of starlike harmonic functions. C R Math Acad Sci Paris, 2016, 354:13-18
[13] Dziok J, Srivastava H M. Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transform Spec Funct, 2003, 14:7-18
[14] Jahangiri J M. Harmonic functions starlike in the unit disk. J Math Anal Appl, 1999, 235:470-477
[15] Jahangiri J M, Silverman H. Harmonic univalent functions with varying arguments. Int J Appl Math, 2002, 8:267-275
[16] Janowski W. Some extremal problems for certain families of analytic functions I. Ann Polon Math, 1973, 28:297-326
[17] Karpuzoullari S Y, Öztürk M, Yamankaradeniz M. A subclass of harmonic univalent functions with negative coefficients. Appl Math Comput, 2003, 142(2/3):469-476
[18] Krein M, Milman D. On the extreme points of regularly convex sets. Studia Mathematica, 1940, 9:133-138
[19] Lewy H. On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull Amer Math Soc, 1936, 42:689-692
[20] Montel P. Sur les families de functions analytiques qui admettent des valeurs exceptionelles dans un domaine. Ann Sci Ecole Norm Sup, 1912, 23:487-535
[21] Mostafa A O, Aouf M G. Goodman-Rřnning-type multivalent harmonic functions based on Dziok-Srivastava operator. Southeast Asian Bull Math, 2015, 39:829-840
[22] Murugusundaramoorthy G, Vijaya K, Raina R K. A subclass of harmonic functions with varying arguments defined by Dziok-Srivastava operator. Arch Math (Brno), 2009, 45:37-46
[23] Omar R, Halim S A. Multivalent harmonic functions defined by Dziok-Srivastava operator. Bull Malays Math Sci Soc, 2012, 35(2):601-610
[24] Öztürk M, Yalçin S, Yamankaradeniz M. Convex subclass of harmonic starlike functions. Appl Math Comput, 2004, 154:449-459
[25] Pathak A L, Dixit K K, Agarwal R. A new subclass of harmonic univalent functions associated with Dziok-Srivastava operator. Int J Math Math Sci, 2012, Art. ID 416875, 10 pp
[26] Silverman H. Harmonic univalent functions with negative coefficients. J Math Anal Appl, 1998, 220:283-289
[27] Xu Q-H, Xiao H-G, Srivastava H M. Some applications of differential subordination and the DziokSrivastava convolution operator. Appl Math Comput, 2014, 230:496-508
[28] Yalçin S, Öztürk M. Harmonic functions starlike of the complex order. Mat Vesnik, 2006, 58:7-11
[29] Yalçin S, Öztürk M, Yamankaradeniz M. On the subclass of Salagean-type harmonic univalent functions. J Inequal Pure Appl Math, 2007, 8:Article 54, 9 pp
