The distribution of zeros and poles of best rational approximants is well understood for the functions $f(x)={|x|}^{\alpha }$, $\alpha >0$. If $f\in C[-1,1]$ is not holomorphic on $[-1,1]$, the distribution of the zeros of best rational approximants is governed by the equilibrium measure of $[-1,1]$ under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, $a$-values, and poles of best real rational approximants of degree at most $n$ to a function $f\in C[-1,1]$ that is real-valued, but not holomorphic on $[-1,1]$. Generalizations to the lower half of the Walsh table are indicated.

The aim of this paper is to discuss the uniqueness of the difference monomials $f^{n}f(z+c)$. It assumed that $f$ and $g$ are transcendental entire functions with finite order and $E_{k)}(1,f^{n}f(z+c))=E_{k)}(1,g^{n}g(z+c))$, where $c$ is a nonzero complex constant and $n$, $k$ are integers. It is proved that if one of the following holds (i) $n\ge 6$ and $k=3$, (ii) $n\ge 7$ and $k=2$, and (iii) $n\ge 10$ and $k=1$, then $fg=t_1$ or $f=t_{2}g$ for some constants $t_2$ and $t_3$ which satisfy $t_2^{n+1}=1$ and $t_3^{n+1}=1$. It is an improvement of the result of Qi, Yang and Liu.

For normalized analytic functions $f\left(z\right)$ with $f\left(z\right)\ne 0$ for , we introduce a univalencecriterion defined by sharp inequality associated with the $n$th derivative of $z/f\left(z\right)$, where $n\in \left\{\mathrm{3,4},5,\dots \right\}$.

For Riemannian manifolds $M$ and $N$, admitting a submersion $\varphi$ with compact fibres, we introduce the projection of a function via its decomposition intohorizontal and vertical components. By comparing the Laplacians on $M$ and $N$, we determine conditions under which a harmonic function on $U={\varphi }^{-1}\left(V\right)\subset M$ projects down, via its horizontal component, to a harmonic function on $V\subset N$.

By introducing the concept of L-limited sets and then L-limited Banach spaces, we obtain some characterizations of it with respect to some well-known geometric properties of Banach spaces, such as Grothendieck property, Gelfand-Phillips property, and reciprocal Dunford-Pettis property. Some complementability of operators on such Banach spaces are also investigated.

We consider the Hyers-Ulam stability for the following fractional differential equations in sense of Srivastava-Owa fractional operators (derivative and integral) defined in the unit disk: , in a complex Banach space. Furthermore, a generalization of the admissible functions in complex Banach spaces is imposed, and applications are illustrated.

We present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a circular slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized, and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.

We present an algorithm for $C^1$ Hermite interpolationusing Möbius transformations of planar polynomial Pythagoreanhodograph(PH) cubics. In general, with PH cubics, we cannotsolve $C^1$ Hermite interpolation problems, since their lack of parametersmakes the problems overdetermined. In this paper, weshow that, for each Möbius transformation, we can introduce anextra parameter determined by the transformation, with which wecan reduce them to the problems determining PH cubics in thecomplex plane $ℂ$. Möbius transformations preserve the PH propertyof PH curves and are biholomorphic. Thus the interpolantsobtained by this algorithm are also PH and preserve the topologyof PH cubics. We present a condition to be met by a Hermitedataset, in order for the corresponding interpolant to be simple orto be a loop. We demonstrate the improved stability of these newinterpolants compared with PH quintics.

We introduce a class of complex-valued biharmonic mappings, denoted by $B{H}^0({\varphi }_k;\sigma ,a,b)$, together with its subclass $TB{H}^0({\varphi }_k;\sigma ,a,b)$, and then generalize the discussions in Ali et al. (2010) to the setting of $B{H}^0({\varphi }_k;\sigma ,a,b)$ and $TB{H}^0({\varphi }_k;\sigma ,a,b)$ in a unified way.

We present a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. For each canonical region, three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the modified Neumann kernels and the adjoint generalized Neumann kernels.

This paper surveys recent advances on univalent logharmonic mappings defined on a simply or multiply connected domain. Topics discussed include mapping theorems, logharmonic automorphisms, univalent logharmonic extensions onto the unit disc or the annulus, univalent logharmonic exterior mappings, and univalent logharmonic ring mappings. Logharmonic polynomials are also discussed, along with several important subclasses of logharmonic mappings.

A generalized Möbius transform is presented. It is based on Dirichlet characters. A general algorithm is developed to compute the inverse $Z$ transform on the unit circle, and an error estimate is given for the truncated series representation.