Several information on the beginning of functional analysis as an important and powerful chapter of mathematics, on the results and people, are given.

‎For a positive integer $n$‎, ‎applying Schwarz's lemma related to analytic functions $w(z)=c_n z^n+\cdots$ in the open unit disk $\mathbb{U}$‎, ‎some assertion in a certain lemma which is well-known as Jack's lemma proven by Miller and Mocanu [J‎. ‎Math‎. ‎Anal‎. ‎Appl‎. ‎65 (1978)‎, ‎289-305] is given‎. ‎Further‎, ‎by using a certain method of the proof of subordination relation which was discussed by Suffridge [Duke Math‎. ‎J‎. ‎37 (1970)‎, ‎775--777] and MacGregor [J‎. ‎London Math‎. ‎Soc‎. ‎(2) 9 (1975)‎, ‎530-536]‎, ‎some differential subordination property concerning with the subordination‎ ‎$$‎ ‎p(z)\prec q(z^n)\qquad(z\in\mathbb{U})‎ ‎$$‎ ‎for functions $p(z)=a+a_n z^n+\cdots$ and $q(z)=a+b_1 z+\cdots$ which are analytic in $\mathbb{U}$ is deduced‎, ‎and an extension of some subordination relation is given‎.

‎The present work continues the study of the stability of the functional equations‎ ‎of the type $f(pr‎, ‎qs)‎ + ‎f(ps‎, ‎qr) = f(p,q)\‎, ‎f(r‎, ‎s)$ namely‎ ‎(i) $f(pr‎, ‎qs)‎ + ‎f(ps‎, ‎qr) = g(p,q)\‎, ‎g(r‎, ‎s)$‎, ‎and (ii)‎ ‎$f(pr‎, ‎qs)‎ + ‎f(ps‎, ‎qr) = g(p,q)\‎, ‎h(r‎, ‎s)$‎ ‎for all $p‎, ‎q‎, ‎r‎, ‎s \in G$‎, ‎where $G$ is an abelian group‎. ‎These‎ ‎functional equations arise in the characterization‎ ‎of symmetrically compositive sumform distance measures‎.

We consider locally defined operators of the form $D^{n}\circ K$ where $D$ is the operator of differentiation and $K$ maps the space of continuous functions into the space of $n$-times\ differentiable functions. As a corollary we obtain a characterization of the Volterra operator. Locally defined operators acting in the space of analytic functions are also discussed.

Let $X$ be a Banach space and suppose $Y\subseteq X$ is a Banach space compactly embedded into $X$, and $(a_k)$ is a weakly null sequence of functionals in $X^*$. Then there exists a sequence $\{\varepsilon_n\} \searrow 0$ such that $|a_n(y)| \leq \varepsilon_n \|y\|_Y$ for every $n\in\mathbb{N}$ and every $y\in Y$. We prove this result and we use it for the study of fast decay of Fourier coefficients in $L^p(\mathbb{T})$ and frame coefficients in the Hilbert setting.

Let $E$, $F$ be two Hilbert $C^*$-modules over $C^*$-algebras $A$ and $B$ respectively. In this paper, by the alternative fixed point theorem, we give the Hyers-Ulam-Rassias stability of the equation $$\ip{U(x), U(y)}=\varphi( \ip{x,y})\qquad(x, y\in E),$$ where $U : E\to F$ is a mapping and $\varphi : A\to B$ is an additive map

In this paper, we use the the Riemann-Liouville fractional integral to develop some new results related to the Hermite-Hadamard inequality. Other integral inequalities related to the Minkowsky inequality are also established. Our results have some relationships with [E. Set, M. E. Ozdemir and S.S. Dragomir, J. Inequal. Appl. 2010, Art. ID 148102, 9 pp.] and [L. BougoffaJ. qualPure and ApplMath 7 (2006), no. 2, Article 60, 3 pp.]. Some interested inequalities of these references can be deduced as some special cases.

The spectral properties of the operators inducing linear dynamical systems on Banach spaces determine the asymptotic properties of the systems. In this note an interaction between the spectral mapping properties of operators and stability properties of the corresponding dynamical systems is presented.

In this paper we discuss the concept of $n$-normed spaces. In particular, we show the equality of four different formulas of $n$-norms in a Hilbert space. In addition, we study the notion of bounded $n$-linear functionals on an $n$-normed space and present some results on it.

In this paper we obtain an algorithm towards solving the two-dimensional moment problem. This algorithm gives the necessary and sufficient conditions for the solvability of the moment problem. It is shown that all solutions of the moment problem can be constructed using this algorithm. In a consequence, analogous results are obtained for the complex moment problem.

Let $\A$ be a unital Banach algebra, and denote the spectral radius of $f\in\A$ by $\rho(f)$. If $\A$ is a uniform algebra and $\rho(fh+1)=\rho(gh+1)$ for all $h\in\A$, then it can be shown that $f=g$, a result that also carries in algebras of bounded linear operators on Banach spaces. On the other hand $\rho(fh)=\rho(gh)$ does not imply $f=g$ in any unital algebra, marking a distinction between the polynomials $p(z,w)=zw+1$ and $p(z,w)=zw$. Such results are known as spectral identification lemmas, and in this work we demonstrate first- and second-degree polynomials of two variables that lead to identification via the spectral radius, peripheral spectrum, or full spectrum in uniform algebras and in algebras of bounded linear operators on Banach spaces. The primary usefulness of identification lemmas is to determine the injectivity of a class of mappings that preserve portions of the spectrum, and results corresponding to the given identifications are also presented.

Several results on splitting of extensions of Banach algebras are generalized to the case of (not necessarily commutative, not necessarily unital) rings or topological rings. Detailed proofs of the results are provided.

Stability in terms of two measures for nonlinear differential equations with ``maxima'' is studied. A special type of stability in terms of two measures is defined. The new type of stability generalizes some of the known in the literature. Sufficient conditions for the defined stability are obtained. Cone-valued continuous Lyapunov functions are applied. Method of Razumikhin as well as comparison method for scalar ordinary differential equations have been employed. The usefulness of the introduced definition and the obtained sufficient conditions is illustrated through an example.

It is shown that a) it is possible to define the topology of any topological algebra by a collection of $F$-seminorms, b) every complete locally uniformly absorbent (complete locally $A$-pseudoconvex) Hausdorff algebra is topologically isomorphic to a projective limit of metrizable locally uniformly\break absorbent algebras (respectively, $A$-($k$-normed) algebras, where $k\in(0,1]$ varies, c) every complete locally idempotent (complete locally $m$-pseudoconvex) Hausdorff algebra is topologically isomorphic to a projective limit of locally idempotent Fréchet algebras (respectively, $k$-Banach algebras, where $k\in(0,1]$ varies) and every $m$-algebra is locally $m$-pseudoconvex. Condition for submultiplicativity of $F$-seminorm is given.