×

Subordination results for spiral-like functions associated with the Srivastava-Attiya operator. (English) Zbl 1244.30015

Let \(U\) be the complex unit disc and \(\mathcal A\) the class of functions \(f\) with \(f(z)=z+a_2z^2+\cdots\). For such functions, the authors define the integral operator \[ \mathcal J_{\mu,b}^{m,k}f(z)=z+\sum_{n=2}^{\infty}C_{n}^{m}(b,\mu) a_{n}z^{n}, \] where \(C_{n}^{m}(b,\mu)=\!\left|\left(\frac{1+b}{n+b}\right)^{\mu}\frac{m!(n+k-2)!}{(k-2)!(n+m-1)!}\right|\), \(b\in\mathbb C\setminus\{\mathbb Z_{0}^{-}\}\), \(\mu\in\mathbb C\), \(k\geq 2\) and \(m>-1\). This is a generalization of other integral operators studied by various authors. Further, for \(0\leq \lambda\), \(\gamma<1\) and \(-\pi/2<\eta<\pi/2\), they define the subclass of \(\mathcal A\), denoted by \(\mathcal G_{\mu,b}^{m,k}(\eta,\gamma,\lambda)\), which consists of functions \(f\) satisfying \[ \mathrm{Re} \left(e^{\mathrm{i}\eta}\frac{z(\mathcal J_{\mu,b}^{m,k}f(z))'}{(1-\lambda)\mathcal J_{\mu,b}^{m,k}f(z)+\lambda(\mathcal J_{\mu,b}^{m,k}f(z))'}\right)>\gamma\cos \eta \] for \(z\in U\). Some subclasses are also given and the main result provides a sufficient condition for a function \(f\in\mathcal A\) to be also in \(\mathcal G_{\mu,b}^{m,k}(\eta,\gamma,\lambda)\). Other properties of the class \(\mathcal G_{\mu,b}^{m,k}(\eta,\gamma,\lambda)\) are given in the second theorem and a corollary following from it.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.2307/2007212 · JFM 45.0672.02
[2] DOI: 10.1090/S0002-9947-1969-0232920-2
[3] DOI: 10.1016/j.amc.2004.12.004 · Zbl 1082.11052
[4] DOI: 10.1016/S0022-247X(02)00500-0 · Zbl 1035.30004
[5] DOI: 10.1016/j.jmaa.2004.05.040 · Zbl 1106.11034
[6] DOI: 10.1016/0022-247X(72)90081-9 · Zbl 0246.30031
[7] DOI: 10.1080/10652460600926907 · Zbl 1184.11005
[8] DOI: 10.1006/jmaa.1993.1204 · Zbl 0774.30008
[9] DOI: 10.1090/S0002-9939-1965-0178131-2
[10] DOI: 10.4153/CJM-1967-038-0 · Zbl 0181.08104
[11] DOI: 10.1016/S0096-3003(03)00746-X · Zbl 1078.11054
[12] DOI: 10.1080/10652460600926923 · Zbl 1172.11026
[13] DOI: 10.1016/j.amc.2004.04.031 · Zbl 1093.30005
[14] DOI: 10.1090/S0002-9939-1966-0188423-X
[15] Prajapat J. K., J. Math. Inequal. 3 pp 129– (2009)
[16] DOI: 10.1080/10652460701542074 · Zbl 1130.30003
[17] DOI: 10.1155/S0161171289000797 · Zbl 0688.30009
[18] DOI: 10.1155/S0161171200004634 · Zbl 0963.30015
[19] Spacek L., Cas. Mat. Fys. 62 (2) pp 12– (1932)
[20] DOI: 10.1080/10652460701208577 · Zbl 1112.30007
[21] Srivastava H. M., Series Associated with the Zeta and Related Functions (2001) · Zbl 1014.33001
[22] DOI: 10.1090/S0002-9939-1961-0125214-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.