Convolution Properties of a Class of Analytic Functions
Abstract
In this paper, we introduce a new class $\mathcal{R}^{\alpha}_{m}(h)$ of functions $F=f\ast\psi$, defined in the open unit disc $E$ with $F(0)=F^{\prime}(0)-1=0$ and satisfying the condition
\begin{equation*}
F^{\prime}(z)+\alpha zF^{\prime\prime}(z)=\left(\frac{m}{4}+\frac{1}{2}\right)p_{_{1}}(z)-\left(\frac{m}{4}-\frac{1}{2}\right)p_{_{2}}(z),
\end{equation*}
for $\alpha\geq0,\,m\geq2$ and $p_{_{i}}\prec h,\, i=1,2.$
Several convolution properties of this class are obtained by using the method of differential subordination. Many relevant connections of the findings here with those in earlier works are pointed out as special cases.
References
Afis Saliu and Khalida Inayat Noor, On subclasses of functions with boundary and radius rotations associated with crescent domains, Bulletin of the Korean Mathematical Society 57(6) (2020), 1529-1539. https://doi.org/10.4134/BKMS.B200039
S.M. Aydoğan and F.M. Sakar, Radius of starlikeness of p-valent $lambda$-fractional operator, Applied Mathematics and Computation 357 (2019), 374-378. https://doi.org/10.1016/j.amc.2018.11.067
D. Bshouty, A. Lyzzaik and F. M. Sakar, Harmonic mappings of bounded boundary rotation, Proceedings of the American Mathematical Society 146 (2018), 1113-1121. http://doi.org/10.1090/proc/13796
B.C. Carlson and D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM Journal on Mathematical Analysis 15(4) (1984), 737-745. https://doi.org/10.1137/0515057
H. Guney, F.M. Sakar and S. Aytas, Subordination results on certain subclasses of analytic functions involving generalized differential and integral operators, Mathematica Aeterna 2 (2012), 177-184.
K. Jabeen and Afis Saliu, Properties of functions with bounded rotation associated with limaçon class, Commun. Korean Math. Soc. 37(4) (2022), 995-1007. https://doi.org/10.4134/CKMS.C210273
W. Janowski, Some extremal problems for certain families of analytic functions I, Annales Polonici Mathematici 28(3) (1973), 297-326.
I.B. Jung, Y.C. Kim and H.M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, Journal of Mathematical Analysis and Applications 176(1) (1993), 138-147. https://doi.org/10.1006/jmaa.1993.1204
S. Kanas and A. Wisniowska, Conic domains and starlike functions, Revue Roumaine de Mathématiques Pures et Appliquées 45(4) (2000), 647-658.
S. Kanas and D. Răducanu, Some class of analytic functions related to conic domains, Mathematica Slovaca 64(5) (2014), 1183-1196. https://doi.org/10.2478/s12175-014-0268-9
Y.C. Kim, K.S. Lee and H.M. Srivastava, Some applications of fractional integral operators and Ruscheweyh derivatives, Journal of Mathematical Analysis and Applications 197(2) (1996), 505-517. https://doi.org/10.1006/jmaa.1996.0035
R.J. Libera, Some classes of regular univalent functions, Proceedings of the American Mathematical Society 16(4) (1965), 755-758. https://doi.org/10.1090/s0002-9939-1965-0178131-2
J.L. Liu and H.M. Srivastava, Certain properties of the Dziok-Srivastava operator, Applied Mathematics and Computation 159(2) (2004), 485-493. https://doi.org/10.1016/j.amc.2003.08.133
S.S. Miller and P.T. Mocanu, Differential subordinations, Theory and applications, Marcel Dekker Inc, New York, Basel, 1999.
K.I. Noor, On some univalent integral operators, Journal of Mathematical Analysis and Applications 128(2) (1987), 586-592. https://doi.org/10.1016/0022-247x(87)90208-3
K.I. Noor, Classes of analytic functions defined by Hadamard product, New Zealand Journal of Mathematics 24 (1995), 53-64.
B. Pinchuk, Functions of bounded boundary rotation, Israel Journal of Mathematics 10(1) (1971), 6-16. https://doi.org/10.1007/bf02771515
S. Ruscheweyh, New criteria for univalent functions, Proceedings of the American Mathematical Society 49 (1975), 109-115. https://doi.org/10.1090/s0002-9939-1975-0367176-1
S. Ruscheweyh, Convolutions in Geometric Function Theory, Fundamental Theories of Physics. Séminaire de Mathématiques Supérieures, 1982.
F.M. Sakar and S.M. Aydoğan, Coefficient bounds for certain subclasses of m-fold symmetric bi-univalent functions defined by convolution, Acta Universitatis Apulensis 55 (2018), 11-21. https://doi.org/10.17114/j.aua.2018.55.02
S. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture, Commentarii Mathematici Helvetici 48(1) (1973), 119-135. https://doi.org/10.1007/bf02566116
R. Singh and S. Singh, Convolution properties of a class of starlike functions, Proceedings of the American Mathematical Society 106(1) (1989), 145-152. https://doi.org/10.1090/s0002-9939-1989-0994388-6
H.M. Srivastava, M. Saigo and S. Owa, A class of distortion theorems involving certain operators of fractional calculus, Journal of Mathematical Analysis and Applications 131 (2) (1988), 412-420. https://doi.org/10.1016/0022-247x(88)90215-6
H.M. Srivastava and A.A. Attiya, An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination, Integral Transforms and Special Functions 18(3) (2007), 207-216. https://doi.org/10.1080/10652460701208577
J. Stankiewicz and Z. Stankiewicz, Some applications of the Hadamard convolution in the theory of functions, Ann. Univ. Mariae Curie-Sklodowska 40 (1986), 251-265.
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
