We first show that the Gaussian integral means of $f:ℂ\to ℂ$ (with respect to the area measure ${\mathrm{e}}^{-{\alpha \left|z\right|}^2}\mathrm{d}A\left(z\right)$) is a convex function of $r$ on $(0,\infty )$ when $\alpha \le 0$. We then prove that the weighted integral means $A_{\alpha ,\beta }(f,r)$ and $L_{\alpha ,\beta }(f,r)$ of the mixed area and the mixed length of $f\left(r𝔻\right)$ and $\partial f\left(r𝔻\right)$, respectively, also have the property of convexity in the case of $\alpha \le 0$. Finally, we show with examples that the range $\alpha \le 0$ is the best possible.

A class of convolution operators on spaces of holomorphic functions related to the Hadamard multiplication theorem for power series and generalizing infinite order Euler differential operators is introduced and investigated. Emphasis is placed on questions concerning injectivity, denseness of range and surjectivity of the operators.

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 $P{⁰}_{\beta }=f\in A:Re{f}^{\text{'}}\left(z\right)>\beta ,z\in \Delta$. For λ > 0, suppose that denotes any one of the following classes of functions: $M_{1,\lambda }^{\left(1\right)}=f\in :Rez{\left(z{f}^{\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}>-\lambda ,z\in \Delta$, $M_{1,\lambda }^{\left(2\right)}=f\in :Rez{\left(z²f^{\text{'}\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}>-\lambda ,z\in \Delta$, $M_{1,\lambda }^{\left(3\right)}=f\in :Re1/2{\left(z{\left(z²f^{\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}\right)}^{\text{'}}-1>-\lambda ,z\in \Delta$. The main purpose of this paper is to find conditions on λ and γ so that each f ∈ is in ${}_{\gamma }$ or ${}_{\gamma }$, γ ∈ [0,1/2]. Here ${}_{\gamma }$ and ${}_{\gamma }$ 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) $-z(a+z)/(1+az)$ is CHD (convex in the horizontal direction) provided [...] a=1 $a=1$ or [...] −1≤a≤0 $-1\le a\le 0$ . 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 ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special...