Two estimates useful in applications are proved for the Fourier transform in the space $\rm{L}^2(\rm{X})$, where $\rm{X}$ a symmetric space, as applied to some classes of functions characterized by a generalized modulus of continuity.

In the present paper, the authors introduce a new subclass of $p$-valent analytic functions with complex order defined on the open unit disk $\mathbb{U}=\{z:z∈\mathbb{C}\ \text{and}\ |z|<1\}$ and obtain coefficient inequalities for the functions in these classes. Applications of these results for the functions defined by the convolution are also obtained.

Let $B(E, F)$ be the set of all bounded linear operators from a Banach space $E$ into another Banach space $F$, $B^+(E, F)$ the set of all double splitting operators in $B(E, F)$ and $GI(A)$ the set of generalized inverses of $A\in B^+(E, F)$. In this paper we introduce an unbounded domain $\Omega(A, A^+)$ in $B(E, F)$ for $A\in B^+(E, F)$ and $A^+\in GI(A)$, and provide a necessary and sufficient condition for $T\in \Omega(A, A^+)$. Then several conditions equivalent to the following property are proved: $B=A^+(I_F+(T−A)A^+)^{−1}$ is the generalized inverse of $T$ with $R(B)=R(A^+)$ and $N(B)=N(A^+)$, for $T\in \Omega(A, A^+)$, where $I_F$ is the identity on $F$. Also we obtain the smooth $(C^∞)$ diffeomorphism $M_A(A^+, T)$ from $\Omega(A, A^+)$ onto itself with the fixed point $A$. Let $S =\{T\in \Omega(A, A^+): R(T)∩N(A^+) = \{0\}\}$, $M(X) =\{T\in B(E, F): TN(X) ⊂ R(X)\}$ for $X ∈ B(E, \mathcal{F})$, and $\mathcal{F} = \{M(X): ∀X\in B(E, F)\}$. Using the diffeomorphism $M_A(A^+, T)$ we prove the following theorem: $S$ is a smooth submanifold in $B(E, F)$ and tangent to $M(X)$ at any $X\in S$. The theorem expands the smooth integrability of $\mathcal{F}$ at $A$ from a local neighborhoold at $A$ to the global unbounded domain $\Omega(A, A^+)$. It seems to be useful for developing global analysis and geomatrical method in differential equations.

In this article, we introduce a notion of nonuniform wavelet frames on local fields of positive characteristic. Furthermore, we gave a complete characterization of tight nonuniform wavelet frames on local fields of positive characteristic via Fourier transform. Our results also hold for the Cantor dyadic group and the Vilenkin groups as they are local fields of positive characteristic.

In this article, the 2-variable general polynomials are taken as base with Peters polynomials to introduce a family of 2-variable Peters mixed type polynomials. These polynomials are framed within the context of monomiality principle and their properties are established. Certain summation formulae for these polynomials are also derived. Examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored.

The objective of this note is to provide some (potentially useful) integral transforms (for example, Euler, Laplace, Whittaker etc.) associated with the generalized $k$-Bessel function defined by Saiful and Nisar [3]. We have also discussed some other transforms as special cases of our main results.

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|≤k$, $k≤1$, then for every real or complex number $β$, with $|β|≤1$ and $R≥1$, it was shown by A. Zireh et al. [7] that for $|z|=1$,
$$\min\limits_{|z|=1}\left|P(Rz)+\beta(\frac{R+k}{1+k})^nP(z)\right|\geq k^{-n}\left|R^n+\beta(\frac{R+k}{1+k})^n\right|\min\limits_{|z|=k}|P(z)|.$$
In this paper, we shall present a refinement of the above inequality. Besides, we shall also generalize some well-known results.

A loopless graph on $n$ vertices in which vertices are connected at least by $a$ and at most by $b$ edges is called a $(a,b,n)$-graph. A $(b,b,n)$-graph is called $(b,n)$-graph and is denoted by $K^b_n$ (it is a complete graph), its complement by $\overline{K}^b_n$. A non increasing sequence $π = (d_1,···,d_n)$ of nonnegative integers is said to be $(a,b,n)$ graphic if it is realizable by an $(a,b,n)$-graph. We say a simple graphic sequence $π = (d_1,···,d_n)$ is potentially $K_4−K_2\cup K_2$-graphic if it has a a realization containing an $K_4−K_2\cup K_2$ as a subgraph where $K_4$ is a complete graph on four vertices and $K_2\cup K_2$ is a set of independent edges. In this paper, we find the smallest degree sum such that every $n$-term graphical sequence contains $K_4−K_2\cup K_2$ as subgraph.