Convolution Conditions for Convexity, Starlikeness and Spiral-Likeness.
Let A denote the space of all analytic functions in the unit disc Δ with the normalization f(0) = f’(0) − 1 = 0. For β < 1, let . For λ > 0, suppose that denotes any one of the following classes of functions: , , . The main purpose of this paper is to find conditions on λ and γ so that each f ∈ is in or , γ ∈ [0,1/2]. Here and respectively denote the class of all starlike functions of order γ and the class of all convex functions of order γ. As a consequence, we obtain a number...
We first prove that the convolution of a normalized right half-plane mapping with another subclass of normalized right half-plane mappings with the dilatation [...] −z(a+z)/(1+az) is CHD (convex in the horizontal direction) provided [...] a=1 or [...] −1≤a≤0 . Secondly, we give a simply method to prove the convolution of two special subclasses of harmonic univalent mappings in the right half-plane is CHD which was proved by Kumar et al. [1, Theorem 2.2]. In addition, we derive the convolution...
For n ∈ ℕ, L > 0, and p ≥ 1 let be the largest possible value of k for which there is a polynomial P ≢ 0 of the form , , , such that divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that . We find the size of and for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special...
Soit une courbe de Jordan fermée rectifiable dans le plan de la variable complexe. On dit que véfifie la condition corde-arc sioù est la longueur du plus petit arc de joignant et . Soit une représentation conforme du disque unité dans l’intérieur de . Nous prouvons que restreint à appartient à la classe de Muckenhoupt et nous en tirons certains corollaires. Dans deux cas particuliers nous montrons que le résultat peut être amélioré.