We show that functions whose derivatives lie in a half-plane are preserved under the Pommerenke, Chandra-Singh, Libera, Alexander and Bernardi integral transforms. We determine precisely how these transforms act on such functions. We prove that if the derivative of a function lies in a convex region then the derivative of its Pommerenke, Chandra-Singh, Libera, Alexander and Bernardi transforms lie in a strictly smaller convex region which can be determined. We also consider iterates of these transforms. Best possible results are obtained.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap91-1-7,
author = {Johnny E. Brown},
title = {Integral transforms of functions with restricted derivatives},
journal = {Annales Polonici Mathematici},
volume = {92},
year = {2007},
pages = {85-97},
zbl = {1129.30005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap91-1-7}
}
Johnny E. Brown. Integral transforms of functions with restricted derivatives. Annales Polonici Mathematici, Tome 92 (2007) pp. 85-97. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap91-1-7/