Improved Bohr’s inequality for locally univalent harmonic mappings

We prove several improved versions of Bohr’s inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $\left|g^{\prime }\left(z\right)\right|\le \left|h^{\prime }\left(z\right)\right|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.     

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk ${\mathbb{D}}$ . Each function ${f\in Co(\alpha)}$ maps the unit disk ${\mathbb{D}}$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional ${(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ . In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional ${(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

Riesz-Fejér Inequalities for Harmonic Functions

Potential Analysis - In this article, we prove the Riesz - Fejér inequality for complex-valued harmonic functions in the harmonic Hardy space hp for all p &gt;?1. The result is sharp...     

Doubly charged protonated a ions derived from small peptides

Protonated a(2) and a(3) (therefore doubly charged) ions in which both charges lie on the peptide backbone are formed in collision-induced dissociations of [La(III)(peptide)(CH(3)CN)(m)](3+) complexes. Abundant (a(3)+H)(2+) ions are formed from triproline (PPP) and peptides with a proline residue at the N-terminus; these peptides are the most effective in producing ions of the type (a(2)+H)(2+) and (a(3)+H)(2+). A systematic study of the effect of the location of the proline residue and other residues of aliphatic amino acids on the generation of protonated a ions is reported. Density functional theory calculations at B3LYP/6-311++G(d,p) gave the proton affinity of the a(3) ion derived from PPP to be 167.6 kcal mol(-1), 2.6 kcal mol(-1) higher than that of water. The protonated a(2) ions of diglycine and diproline and a(3) ions of triglycine have lower proton affinities and are only observed in lower abundances, possibly due to proton transfer to water in ion-molecule reactions.     

Asymmetric epoxidation of some arylalkenyl sulfones using a modified Juliá-Colonna procedure

A modified procedure for performing the Juliá-Colonna epoxidation reaction effects the oxidation of some vinyl sulfones to generate the corresponding epoxides 5-8 in good to excellent optical purity.     

Absolutely convex,uniformly starlike and uniformly convex harmonic mappings

In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disc. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi-type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hypergeometric and polylogarithm functions.     

Maximal Area Integral Problem for Certain Class of Univalent Analytic Functions

One of the classical problems concerns the class of analytic functions f on the open unit disk |z| < 1 which have finite Dirichlet integral Δ(1, f), where

$$\Delta(r ,f) = \iint_{|z| < r} |f' (z)| ^ 2 \, {\rm d} x {\rm d}y \quad (0 < r \leq 1)$$

The class \({\mathcal{S} ^*(A,B)}\) of normalized functions f analytic in |z| < 1 and satisfies the subordination condition \({zf'(z)/f(z)\prec (1+Az)/(1+Bz)}\) in |z| < 1 and for some \({-1\leq B\leq 0}\) , \({A \in \mathbb{C}}\) with \({A\neq B}\) , has been studied extensively. In this paper, we solve the extremal problem of determining the value of

$$\max_{f\in \mathcal{S}^*(A,B)}\Delta(r,z/f)$$

as a function of r. This settles the question raised by Ponnusamy and Wirths (Ann Acad Sci Fenn Ser AI Math 39:721–731, 2014). One of the particular cases includes solution to a conjecture of Yamashita which was settled recently by Obradovi? et al. (Comput Methods Funct Theory 13:479–492, 2013).