Abstract

In this paper, we define the subclasses $\mathcal{V_\delta}(A,B)$ and $\mathcal{K_\delta}(A,B)$ of analytic functions by using $\Omega^{\delta}f(z)$. For functions belonging to these classes, we obtain coefficient estimates, distortion bounds and many more properties.

Keywords: Univalent functions; Komato operator; fractional derivative; linear operator.

MSC: 30C45

Abstract

Let $R$ be a commutative ring with identity. Let $\phi: \sI\to \eI$ be a function where $\sI$ denotes the set of all ideals of $R$. Let $I$ be an ideal of $R$. An element $a\in R$ is called $\phi$-prime to $I$ if $ra\in I - \phi(I)$ (with $r\in R$) implies that $r\in I$. We denote by $S_\phi(I)$ the set of all elements of $R$ that are not $\phi$-prime to $I$. $I$ is called a $\phi$-primal ideal of $R$ if the set $P := S_\phi(I)\cup \phi(I)$ forms an ideal of $R$. So if we take $\phi_{\emptyset}(Q) = \emptyset$ (resp., $\phi_0(Q) = 0$), a $\phi$-primal ideal is primal (resp., weakly primal). In this paper we study the properties of several generalizations of primal ideals of $R$.

Keywords: Primal ideal; weakly primal ideal; $\phi$-primal ideal.

MSC: 13A15, 13A10

Abstract

In the present paper, we define a new class of normalized analytic functions based on the $\Phi$-like type. Some inequalities are introduced based on the concept of the subordination in the unit disk. The convexity techniques are used to obtain the main result. Some applications are posed on classes of functions such as starlike functions, convex functions, starlike and convex functions of complex order.

Keywords: Univalent functions; Starlike functions; Convex functions; Differential subordination; Superordination; Unit disk; Starlike and convex functions of complex order; $\Phi$-like functions; Convexity techniques.

MSC: 30C99, 30C45

Abstract

In this paper we deal with special {\it sequence spaces equations (SSE) with operators}, which are determined by an identity whose each term is a {\it sum or a sum of products of sets of the form $\chi_{a}(T)$ and $\chi_{f(x)}(T)$} where $f$ maps $U^{+}$ to itself, and $\chi$ is any of the symbols $s$, $s^{0}$, or $s^{(c)}$. We solve the equation $\chi_{x}(\Delta )=\chi_{b}$ where $\chi$ is any of the symbols $s$, $s^{0}$, or $s^{(c)}$ and determine the solutions of (SSE) with operators of the form $(\chi_{a}\ast\chi_{x}+\chi_{b})(\Delta)=\chi_{\eta}$ and $[\chi_{a}\ast(\chi_{x})^{2}+\chi_{b}\ast\chi_{x}](\Delta)=\chi_{\eta}$ and $\chi_{a}+\chi_{x}(\Delta)=\chi_{x}$ where $\chi$ is any of the symbols $s$, or $s^{0}$.

Keywords: Sequence space; operator of the first difference; BK space; infinite matrix; sequence spaces equations (SSE); (SSE) with operators.

MSC: 40C05, 46A15

Abstract

In this paper we prove that the constants in some inequalities for entire functions of exponential type (trigonometric polynomials) are best possible. Also, we prove an inequality of Bernstein type for entire functions having some additional properties.

Keywords: Bernstein inequality, entire functions of exponential type.

MSC: 41A17, 30D15, 26D05

Abstract

Within this paper, the weaker separation axioms of $T_0$, $T_1$, $R_0$, $T_2$, and $R_1$ are further characterized using mathematical induction, closed sets, convergence, and $T_0$-identification spaces. The results are used to further investigate general topological spaces, to further investigate constant nets and sequences, and finite nets and sequences in topological spaces.

Keywords: $T_0$-space; $T_1$-space; $T_2$-space; $R_0$-space; $R_1$-space.

MSC: 54D10, 54A20

Abstract

In this paper, we establish some relations between the zeros and coefficients of a polynomial and thereby prove a few results concerning stable polynomials.

Keywords: Zeros and coefficients; stable polynomials; inequalities in the complex domain.

MSC: 12D10, 26C10, 26D15