‎‎‎‎‎In this paper‎, ‎we present some interesting results related to the bounds of Zipf-Mandelbrot entropy and the $3$-convexity of the function‎. ‎Further‎, ‎we define linear functionals as the nonnegative differences of the obtained inequalities and we present mean value theorems for the linear functionals‎. ‎Finally‎, ‎we discuss the $n$-exponential convexity and the log-convexity of the functions‎ ‎associated with the linear functionals‎.

‎We introduce a variable exponent version of the Hardy space of analytic functions on the unit disk‎. ‎We then show some properties of the space and give an example of a variable exponent $p(\cdot)$ that satisfies the $\log$-Hölder condition and $H^{p(\cdot)}\neq H^q$ for every constant exponent $q \in (1, \infty)$‎. ‎We also consider a variable exponent version of the Hardy space on the upper-half plane.‎

‎‎‎The Morrey boundedness is proved for the Riesz transform and the inverse operator of the nondegenerate elliptic differential operator of divergence form generated by a vector-function in $(L^\infty)^{n^2}$‎, ‎and for the inverse operator of the Schrödinger operators whose nonnegative potentials satisfy a certain integrability condition‎. ‎In this note‎, ‎our result is not obtained directly from the estimates of integral formula‎, ‎which reflects the fact that the solution of the Kato conjecture did not use any integral expression of the operators‎. ‎One of the important tools in the proof is the decomposition of the functions in Morrey spaces based on the elliptic differential operators in question‎. ‎In some special cases where the integral kernel comes into play‎, ‎the boundedness property of the Littlewood-Paley operator was already obtained by Gong‎. ‎So‎, ‎the main novelties of this paper are the decomposition results associated with elliptic differential operators and the result in the case where the explicit formula of the integral kernel of the heat semigroup is unavailable‎.

‎We introduce the notion of an $m$-convex set-valued function and study some properties of this class of functions‎. ‎Several characterizations are given as well as certain algebraic properties and examples‎. ‎Finally‎, ‎an inclusion of Jensen type is presented jointly with a sandwich type theorem‎.

‎‎‎We introduce near-martingales in the setting of quantum probability spaces and present a trick for investigating some of their properties‎. ‎For instance‎, ‎we give a near-martingale analogous result of the fact that the space of all bounded $L^p$-martingales‎, ‎equipped with the norm $\|\cdot\|_p$‎, ‎is isometric to $L^p(\mathfrak{M})$ for $p>1$‎. ‎We also present Doob and Riesz decompositions for the near-submartingale and provide Gundy's decomposition for $L^1$-bounded near-martingales‎. ‎In addition‎, ‎the interrelation between near-martingales and the instantly independence is studied‎.

‎For a class of continuous functions including complex polynomials in $z$ and $\bar{z},$ we show that‎ ‎the corresponding Toeplitz operator on the Bergman space of the unit disk‎ ‎can be expressed as a quotient of certain differential operators with holomorphic coefficients‎. ‎This enables us to obtain several nontrivial operator theoretic results about such Toeplitz operators‎, ‎including a new criterion for invertibility of a Toeplitz operator for a class of harmonic symbols‎. ‎

‎‎We study a new class of fractional neutral differential control system with noninstantaneous impulses and state-dependent delay‎. ‎The resolvent family and Krasnoselskii's fixed point theorem are utilized to examine the approximate controllability outcomes for the proposed system‎. ‎Further‎, ‎we derive the trajectory controllability outcomes for the proposed fractional control system‎. ‎Finally‎, ‎the main results are validated with the aid of an example‎.

‎‎Using the three critical points theorem‎, ‎we obtain the existence of three weak solutions for a Kirchhoff-type problem involving the nonlocal fractional $p$-Laplacian operator in a fractional Sobolev space‎, ‎with homogeneous Dirichlet boundary conditions‎.

We build a process in order to extend the truncated weighted shift‎, ‎using techniques of the bi-indexed recursive sequences‎. ‎We apply this process to solve the subnormality of $2$-variable weighted shifts‎, ‎whose associated moment sequence is a bi-indexed recursive sequence‎. ‎Notably‎, ‎we detail the case of the truncated $2$-variable weighted shift $T\equiv(T_1‎, ‎T_2)$ of order $(2,2)$‎.

‎‎‎‎Using ultrapowers of $C^*$-algebras we provide a new construction of the multiplier algebra of a $C^*$-algebra‎. ‎This extends the work of Avsec and Goldbring [Houston J‎. ‎Math.‎, ‎to appear‎, ‎arXiv:1610.09276]‎ ‎to the setting of noncommutative and non separable $C^*$-algebras‎. ‎We also extend their work‎ ‎to give a new proof of the fact that groups acting transitively on locally finite trees with boundary amenable stabilizers are‎ ‎boundary amenable‎.