Volterra-type operator on the subclasses of univalent functions

Keywords: operators, convex functions of complex order, starlike functions of complex order, spirallike functions of type λ with complex order, Schwazian derivative, Schwazian norm, composition operator, Volterra type operator

Abstract

UDC 517.5

In this article, we examine the necessary and sufficient conditions for a member to belong to the class of starlike and convex functions of complex order $b$ $(b\neq 0)$ and spirallike functions of type $ \lambda$ $\Big( {-\dfrac{\pi}{2}}<\lambda<\dfrac{\pi}{2} \Big)$ with the complex order $b$ $(b\neq 0).$
We obtain sharp estimates for the coefficient of the second term in the Taylor series of functions belonging to the mentioned classes.

In the main part of this paper, we obtain the necessary and sufficient conditions of boundedness for the image of the open unit disk $ \mathbb{D}=\lbrace z\in \mathbb{C}:\vert z\vert<1\rbrace $ under the action of a Volterra-type operator and the product of the composition operator and Volterra-type operator in the space of univalent functions and its subspace.
Finally, we obtain an estimate of the Schwartzian norm of the above operators in these spaces.

References

A. K. Bakhtin, I. V. Denega, Inequalities for the inner radii of nonoverlapping domains, Ukr. Math. J., 71, № 7, 1138 – 1145 (2019). DOI: https://doi.org/10.1007/s11253-019-01703-x

A. K. Bakhtin, G. P. Bakhtina, Yu. B. Zelinskii, Topological-algebraic structures and geometric methods in complex analysis (in Russian), Inst. Math. NAS Ukraine, Kiev (2008).

C. C. Cowen, B. D. Maccluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL (1994).

I. Graham, G. Kohr, Geometric function theory in one and higher dimensions, Marcel Dekker Inc., New York (2003), https://doi.org/10.1201/9780203911624 DOI: https://doi.org/10.1201/9780203911624

Z. J. Hu, Extended Ces´aro operators on mixed normed spaces, Proc. Amer. Math. Soc., 131, 2171 – 2179 (2003), https://doi.org/10.1090/S0002-9939-02-06777-1 DOI: https://doi.org/10.1090/S0002-9939-02-06777-1

R. Kargar, Volterra-type integral operator on analytic function spaces; https://arxiv.org/pdf/1805.01043.pdf

E. G. Kwon, J. Lee, Essential norm of the composition operators between Bergman spaces of logarithmic weights, Bull. Korean Math. Soc., 54, № 1, 187 – 198 (2017), https://doi.org/10.4134/BKMS.b160014 DOI: https://doi.org/10.4134/BKMS.b160014

E. G. Kwon, J. Lee, Composition operator between Bergman spaces of logarithmic weights, Internat. J. Math. Oper., 26, № 9, Article 1550068 (2015), 14 p., https://doi.org/10.1142/S0129167X15500688 DOI: https://doi.org/10.1142/S0129167X15500688

S. Li, S. Stevi´c, Products of integral-type operators and composition operators between Bloch-type spaces, J. Math. Anal. and Appl., 349, 596 – 610 (2009), https://doi.org/10.1016/j.jmaa.2008.09.014 DOI: https://doi.org/10.1016/j.jmaa.2008.09.014

S. Li, S. Stevi´c, Products of Volterra-type operator and composition operator from $H^{infty}$ and Bloch space to the Zygmund space, J. Math. Anal. and Appl., 345, № 1, 40 – 52 (2008), https://doi.org/10.1016/j.jmaa.2008.03.063 DOI: https://doi.org/10.1016/j.jmaa.2008.03.063

S. Li, S. Stevi´c, Products of composition and integral type operators from $H^{infty}$ to the Bloch space, Complex Var. and Elliptic Equat., 53, № 5, 463 – 474 (2008), https://doi.org/10.1080/17476930701754118 DOI: https://doi.org/10.1080/17476930701754118

Z. Nehari, Aproperty of convex conformal maps, J. Anal. Math., 30, 390 – 393 (1976), https://doi.org/10.1007/BF02786725 DOI: https://doi.org/10.1007/BF02786725

S. Owa, H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Can. J. Math., 39, 1057 – 1077 (1987), https://doi.org/10.4153/CJM-1987-054-3 DOI: https://doi.org/10.4153/CJM-1987-054-3

R. Yoneda, Pointwise multipliers from BMOA$^α$ to BMOA$^β$ , Complex Var., 49, № 14, 1045 – 1061 (2004), https://doi.org/10.1080/02781070412331320448 DOI: https://doi.org/10.1080/02781070412331320448

Published
24.01.2022
How to Cite
Mahboobi, M. “Volterra-Type Operator on the Subclasses of Univalent Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 1, Jan. 2022, pp. 77 -88, doi:10.37863/umzh.v74i1.1116.
Section
Research articles