Document Type: Research Paper

Authors

Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.

Abstract

‎In this note, we study the integral operators $I_{g}^{\gamma, \alpha}$ and $J_{g}^{\gamma, \alpha}$ of an analytic function $g$ on convex and starlike functions of a complex order. Then, we investigate the same operators on $H^{\infty}$ and Besov spaces.

Keywords

[1] J.J. Donaire, D. Girela, and D. Vukotic, On the growth and range of functions in Mobius invariant spaces, J. Anal. Math., 112(1) (2010), pp. 237-260.

[2] J.J. Donaire, D. Girela, and D. Vukotic, On univalent functions in some Mobius invariant spaces, J. Reine. Angew. Math., 553 (2002), pp. 43-72.

[3] A. Ebadian and J. Sokol, Volterra type operator on the convex functions, Hacet. J. Math. Stat., 47(1) (2018), pp. 57-67.

[4] C. Hammond, The norm of a composition operator with linear symbol acting on the Dirichlet space, J. Math. Anal. Appl., 303(2) (2005), pp. 499-508.

[5] Y.C. Kim and T. Sugawa, Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Rocky Mt. J. Math., 32 (2002), pp. 179-200.

[6] Y.C. Kim and T. Sugawa, Uniformly locally univalent functions and Hardy spaces, J. Math. Anal. Appl., 353(1) (2009), pp. 61-67.

[7] S. Li, Volterra composition operators between weighted bergman spaces and bloch type spaces, J. Korean Math. SOC., 45(1) (2008), pp. 229-248.

[8] S. Li and S. Stevic, Integral type operators from mixed-norm spaces to $alpha$-Bloch spaces, Integr. Transf. Spec. F., 18(7) (2007), pp. 485-493.

[9] S. Li and S. Stevic, Products of integral-type operators and composition operators between bloch-type spaces, J. Math. Anal. Appl., 349(2) (2009), pp. 596-610.

[10] Z. Nehari, A property of convex conformal maps, J. Anal. Math., 30(1) (1976), pp. 390-393.

[11] Z. Orouji and R. Aghalary, The norm estimates of pre-schwarzian derivatives of spirallike functions and uniformly convex $ alpha $-spirallike functions, Sahand Commun. Math. Anal., 12(1) (2018), pp. 89-96.

[12] M. Taati, S. Moradi, and S. Najafzadeh, Some properties and results for certain subclasses of starlike and convex functions, Sahand Commun. Math. Anal.,7(1) (2017), pp. 1-15.

[13] J. Xiao, Holomorphic $Q$ classes, Lecture notes in mathematics, 2001.

[14] K. Zhu, Operator theory in function spaces, MR 92c, 47031, 1990.