Given a function $f\left(t\right)\ge 0$ on $ℝ$ with ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2\left)\right)dt\<\infty$ and $\left|f\right(t)-f(t^{\text{'}}\left)\right|\le l|t-t^{\text{'}}|$, a procedure is exhibited for obtaining on $ℂ$ a (finite) superharmonic majorant of the function$$F\left(z\right):\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{\left|𝔍z\right|}{|z-t{|}^2}f\left(t\right)dt-Al\left|𝔍z\right|,$$where $A$ is a certain (large) absolute constant. This leads to fairly constructive proofs of the two main multiplier theorems of Beurling and Malliavin. The principal tool used is a version of the following lemma going back almost surely to Beurling: suppose that $f\left(t\right)$, positive and bounded away from 0 on $ℝ$, is such that ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2)dt\<\infty$ and denote, for any constant $\alpha \>0$ and each $x\in ℝ$, the unique...

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...

The $k$-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations $F_k\left[u\right]=0$, where $F_k\left[u\right]$ is the elementary symmetric function of order $k$, $1\le k\le n$, of the eigenvalues of the Hessian matrix $D^{2}u$. For example, $F_1\left[u\right]$ is the Laplacian $\Delta u$ and $F_n\left[u\right]$ is the real Monge-Ampère operator det $D^{2}u$, while $1$-convex functions and $n$-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative $k$-convex functions, and give several...