A continuous complex-valued function F in a domain $$D\subseteq \mathbf {C}$$ is poly-analytic of order $$\alpha $$ if it satisfies $$\partial ^{\alpha }F/\partial \overline{z}^{\alpha }=0$$ . One can show that has the form $$F(z)=\sum _{k=0}^{\alpha -1}\overline{z}^{k}A_{k}(z)$$ , where each $$A_k$$ an analytic function. In this paper, we prove existence Landau constant for functions and ...

The purpose of this paper is to prove several results in approximation by complex Picard, Poisson-Cauchy, and Gauss-Weierstrass singular integrals with Jackson-type rate, having the quality of preservation of some properties in geometric function theory, like the preservation of coefficients’ bounds, positive real part, bounded turn, starlikeness, and convexity. Also, some sufficient conditions...

Via a unified geometric approach, class of generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they univalent. Some consequences deduced concerning radius convergence for the Maclaurin series, commutation rotation, continuation beyond domain univalence, and periodicity.

In the present paper, using Jack’s lemma, the authors investigate the differential inequality ∣∣∣(1 − α) Ip(n,λ)f (z) zp + α Ip(n + 1, λ)f (z) Ip(n,λ)f (z) − 1 ∣∣∣ < μ, z ∈ E regarding multivalent functions defined by multiplier transfomation in the open unit disk E = {z : |z| < 1} . As consequences, sufficient conditions for univalence, starlikeness and strongly starlikeness of certain analyti...

The purpose of this paper is to prove several results in approximations through complex convolution polynomials with Jackson-type rate or with best approximation rate, having the quality of preservation of some properties in geometric function theory, like the preservation of: coefficients’ bounds, positive real part, bounded turn, close-to-convexity, starlikeness, convexity, spirallikeness, α-...