### A Subclass of Noor-Type Harmonic p-Valent Functions Based on Hypergeometric Functions

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**Authors:**H. F. AL-JANABY AND F. GHANIM

**DOI:**10.46793/KgJMat2104.499J

**Abstract:**

In this paper, we introduce a new generalized Noor-type operator of harmonic p-valent functions associated with the Fox-Wright generalized hypergeometric functions (FWGH-functions). Furthermore, we consider a new subclass of complex-valued harmonic multivalent functions based on this new operator. Several geometric properties for this subclass are also discussed.

**Keywords:**

Harmonic multivalent function, convolution product, Noor integral operator, Fox-Wright generalized hypergeometric function.

**References:**

[1] O. P. Ahuja and H. Silverman, Inequalities associating hypergeometric functions with planer harmonic mapping, Journal of Inequalities in Pure and Applied Mathematics 5(4) (2004), 1–21.

[2] O. P. Ahuja and J. M. Jahangiri, Multivalent harmonic starlike functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 55 (2001), 1–13.

[3] O. P. Ahuja and P. Sharma, Inclusion theorems involving Wright’s generalized hypergeometric functions and harmonic univalent functions, Acta Univ. Apulensis Math. Inform. 32 (2012), 111–128.

[4] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. Math. 17(1) (1915), 12–22.

[5] H. F. Al-Janaby and F. Ghanim, Third-order diﬀerential Sandwich type outcome involving a certain linear operator on meromorphic multivalent functions, International Journal of Pure and Applied Mathematics 118(3) (2018), 819–835.

[6] H. F. Al-Janaby, F. Ghanim and M. Darus, Third-order diﬀerential Sandwich-type result of meromorphic p-valent functions associated with a certain linear operator, Communications in Applied Analysis 22 (2018), 63–82.

[7] H. F. Al-Janaby and M. Z. Ahmad, Diﬀerential inequalities related to Sǎlǎgean type integral operator involving extended generalized Mittag-Leﬄer function, J. Phys. Conf. Ser. 1132(012061) (2019), 63–82.

[8] H. F. Al-Janaby, On certain of complex harmonic functions involving a diﬀerential operator, Journal of Advanced Research in Dynamical and Control Systems 10 (2018), 27–36.

[9] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429–446.

[10] R. Chandrashekar, G. Murugusundaramoorthy, S. K. Lee and K. G. Subramanian, A class of complex valued harmonic functions deﬁned by Dzoik Srivastava operator, Chamchuri J. Math. 1(2) (2009), 31–42.

[11] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Math. 9 (1984), 3–25.

[12] L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154(1-2) (1984), 137–152.

[13] E. Deniz, On the univalence of two general integral operator, Filomat 29(7) (2015), 1581–1586.

[14] J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103(1) (1999), 1–13.

[15] J. Dziok and R. K. Raina, Families of analytic functions associated with the Wright generalized hypergeometric function, Demonstr. Math. 37(3) (2004), 533–542.

[16] R. M. El-Ashwah and M. K. Aouf, New classes of p-valent harmonic functions, Bull. Math. Anal. Appl. 2(3) (2010), 53–64.

[17] R. M. El-Ashwah, M. K. Aouf and S. M. El-Deeb, On integral operator for certain classes of p-valent functions associated with generalized multiplier transformations, J. Egyptian Math. Soc. 22 (2014), 31–35.

[18] R. M. El-Ashwah and A. H. Hassan, Third-order diﬀerential subordination and superordination results by using Fox-Wright generalized hypergeometric function, Functional Analysis: Theory, Method & Applications 2 (2016), 34–51.

[19] C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc. 27(2) (1928), 389–400.

[20] B. A. Frasin, Univalency of general integral operator deﬁned by Schwarz functions, J. Egyptian Math. Soc. 21 (2013), 119–122.

[21] B. A. Frasin and V. Breaz, Univalence conditions of general integral operator, Mat. Vesnik 65(3) (2013), 394–402.

[22] J. Hadamard, Théorème sur les séries entières, Acta Math. 22 (1899), 55–63.

[23] S. Hussain, A. Rasheed and M. Darus, A subclass of harmonic functions related to a convolution operator, J. Funct. Spaces 2016 (2016), 1–6.

[24] R. W. Ibrahim and M. Darus, New classes of analytic functions involving generalized Noor integral operator, J. Inequal. Appl. 2008 (2008), 1–14.

[25] A. A. Kilbas, M. Saigo and J. J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal. 5 (2002), 437–460.

[26] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755–758.

[27] S. S. Miller, P. T. Mocanu and M. O. Reade, Bazilevic functions and generalized convexity, Rev. Roumaine Math. Pures Appl. 19 (1974), 213–224.

[28] S. S. Miller, P. T. Mocanu and M. O. Reade, Starlike integral operators, Paciﬁc J. Math. 79 (1978), 157–168.

[29] A. O. Mostafa, Some classes of multivalent harmonic functions deﬁned by convolution, Electron. J. Math. Anal. Appl. 2(1) (2014), 246–255.

[30] G. Murugusundaramoorthy and R. K. Raina, On a subclass of harmonic functions associated with the Wright’s generalized hypergeometric functions, Hacet. J. Math. Stat. 38(2) (2009), 129–136.

[31] K. L. Noor, On new classes of integral operators, Journal of Natural Geometry 16 (1999), 71–80.

[32] K. L. Noor, Integral operators deﬁned by convolution with hypergeometric functions, Appl. Math. Comput. 182(2) (2006), 1872–1881.

[33] K. W. Ong, S. L. Tan and Y. E. Tu, Integral operators and univalent functions, Tamkang J. Math. 43(2) (2012), 215–221.

[34] N. N. Pascu and V. Pescar, On integral operators of Kim-Merkes and Pfaltzgraﬀ, Stud. Univ. Babeş-Bolyai Math. 32(55) (1990), 185-192.

[35] S. Rahrovi, On a certain subclass of analytic univalent function deﬁned by using Komatu integral operator, Stud. Univ. Babeş-Bolyai Math. 61(1) (2016), 27–36.

[36] R. K. Raina and P. Sharma, Harmonic univalent functions associated with Wright’s generalized hypergeometric function, Integral Transforms Spec. Funct. 22 (2011), 561–572.

[37] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115.

[38] T. M. Seoudy, On a linear combination of classes of harmonic p-valent functions deﬁned by certain modiﬁed operator, Bull. Iranian Math. Soc. 40(6) (2014), 1539–1551.

[39] P. Sharma, Some Wgh inequalities for univalent harmonic analytic functions, Appl. Math. 1(6) (2010), 464–469.

[40] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. Lond. Math. Soc. 10 (1935), 286–293.

[41] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. Lond. Math. Soc. 46(2) (1940), 389–408.

[42] E. Yaşar and S. Yalçin, Properties of a subclass of multivalent harmonic functions deﬁned by a linear operator, General Mathematics Notes 13(1) (2012), 10–20.