Let $\Sigma$ denote the class of functions $$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$$ belonging to the normalized analytic function class $\mathcal{A}$ in the open unit disk $\mathbb{U}$, which are bi-univalent in $\mathbb{U}$, that is, both the function $f$ and its inverse $f^{-1}$ are univalent in $\mathbb{U}$. The usual method for computation of the coefficients of the inverse function $f^{-1}(z)$ by means of the relation $f^{-1}\big(f(z)\big)=z$ is too difficult to apply in the case of $m$-fold symmetric analytic functions in $\mathbb{U}$. Here, in our present investigation, we aim at overcoming this difficulty by using a general formula to compute the coefficients of $f^{-1}(z)$ in conjunction with the residue calculus. As an application, we introduce two new subclasses of the bi-univalent function class $\Sigma$ in which both $f(z)$ and $f^{-1}(z)$ are $m$-fold symmetric analytic functions with their derivatives in the class $\mathcal{P}$ of analytic functions with positive real part in $\mathbb{U}$. For functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for $|a_{m+1}|$ and $|a_{2m+1}|$.

This paper deals with the derivation of certain new Pólya-Szegő type inequalities by making use of the Saigo fractional integral operator. The results obtained cover the same kind of conclusions in the case of Riemann-Liouville and Erdélyi-Kober fractional integral operators.

In this paper, vector norm inequalities that provides upper bounds for the Lipschitz quantity $\left\Vert f\left( T\right) x-f\left( V\right)x\right\Vert $ for power series $f\left( z\right) =\sum_{n=0}^{\infty}a_{n}z^{n},$ bounded linear operators $T,V$ on the Hilbert space $H$ and vectors $x\in H$ are established. Applications in relation to Hermite-Hadamard type inequalities and examples for elementary functions of interest are given as well.

The Bessel functions $J_{\nu}(x)$ and their derivatives $J_{\nu}^{\prime}(x)$ can be represented by infinite series and infinite products. Using these representations we give very simple proofs for known results concerning the zeros of the above functions.

In this paper, we have established Hermite-Hadamard type inequalities for functions whose 3rd derivatives are s-convex depending on a parameter. These results have generalized some relationships with [4].

In this paper, we introduce the notation of $\varphi -$convex functions on the co-ordinates and present some properties. Also, new Hadamard type inequalities for such functions are obtained.

In this paper some new inequalities are proved related to left hand side of Hermite-Hadamard inequality for the classes of functions whose derivatives of absolute values are $(\alpha ,m)-$convex.

In the paper, the equivalence of the functional inequality $$\|2f(x)+f(y)+f(-y)-f(x-y)\|\leq\|f(x+y)\|\;\;\;(x,y\in{G})$$ and the Drygas functional equation $$f(x+y)+f(x-y)=2f(x)+f(y)+f(-y)\;\;\;(x,y\in{G})$$ is proved for functions $f:G\rightarrow E$ where $(G, +)$ is an abelian group, $(E, \lt\cdot, \cdot\gt)$ is an inner product space, and the norm is derived from the inner product in the usual way.

For analytic function of the form $f_i(z)=z^p+\sum_{n=2}a_n^iz^n$, in the open unit disk, a class $\Gamma_\alpha^p(C_1,C_2;\gamma)$ is introduced and some properties for $\Gamma_\alpha^p(C_1,C_2;\gamma)$ of $f_i(z)$ in relation to coefficient bounds, convex conbination and convolution were obtained.

In this study, we define $g-$convex dominated functions on the co-ordinates and prove some Hadamard-type, Fejer-type inequalities for this new class of functions. We also give some results related to the functional $H$.

Using the fixed point theorem we establish the Hyers-Ulam-Rassias stability of the generalized Pexider functional equation $$\frac{1}{\mid K\mid}\sum_{k\in K}f(x+k\cdot y)=g(x)+h(y),\;\;x,y\in E$$ from a normed space $E$ into a complete $\beta$-normed space $F$, where $K$ is a finite abelian subgroup of the automorphism group of the group $(E,+)$.

The famous Tricomi equation was established in 1923, by F. G. Tricomi, who is the pioneer of parabolic elliptic and hyperbolic boundary value problems and related problems of variable type. In 1945, F. I. Frankl established a generalization of these problems for the well-known Chaplygin equation. In 1953 and 1955, M. H. Protter generalized these problems even further. In 1977, we generalized these results in several n-dimensional simply connected domains. In 1950-1951, M. A. Lavrentjev and A. V. Bitsadze investigated the Bitsadze - Lavrentjev equation. In 1990, we proposed the exterior Tricomi problem. In 2002, we considered uniqueness of quasi-regular solutions for a bi-parabolic elliptic bi-hyperbolic Tricomi problem. In 2006, G. C. Wen investigated the exterior Tricomi problem for general mixed type equations. In 2011, we established the exterior Tricomi and Frankl problems for quaterelliptic - quaterhyperbolic equations. In 2014, D. Amanov and J. M. Rassias investigated boundary value problems for the higher order generalized mixed-parabolic equation. In this paper we investigate the exterior Bitsadze-Lavrentjev problem for quaterelliptic -quaterhyperbolic Bitsadze-Lavrentjev PDEquations with eight parabolic lines in a doubly connected domain and propose open problems. These problems are of vital importance in fluid mechanics.