Let D = z: Re z < 0 and let S*(D) be the class of univalent functions normalized by the conditions , a a finite complex number, 0 ∉ f(D), and mapping D onto a domain f(D) starlike with respect to the exterior point w = 0. Some estimates for |f(z)| in the class S*(D) are derived. An integral formula for f is also given.
In this paper we discuss some subordination results for a subclass of functions analytic in the unit disk U
In this paper, we obtain new sufficient conditions for the operators Fα1,α2,…,αn,β(z) and Gα1,α2,…,αn,β(z) to be univalent in the open unit disc U, where the functions f1, f2, …, fn belong to the classes S*(a, b) and K(a, b). The order of convexity for the operators Fα1,α2,…,αn,β(z) and Gα1,α2,…,αn,β(z) is also determined. Furthermore, and for β = 1, we obtain sufficient conditions for the operators Fn(z) and Gn(z) to be in the class K(a, b). Several corollaries and consequences of the main results...
We geometrically define subclasses of starlike functions related to the class of uniformly starlike functions introduced by A. W. Goodman in 1991. We give an analytic characterization of these classes, some radius properties, and examples of functions in these classes. Our classes generalize the class of uniformly starlike functions, and many results of Goodman are special cases of our results.
We prove two Dyakonov type theorems which relate the modulus of continuity of a function on the unit disc with the modulus of continuity of its absolute value. The methods we use are quite elementary, they cover the case of functions which are quasiregular and harmonic, briefly hqr, in the unit disc.