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Consider the following class of conformable time-fractional stochastic equation $$T_{\alpha,t}^a u(x,t)=\lambda\sigma(u(x,t))\dot{W}_t,\,\,\,\,x\in\mathbb{R},\,t\in[a,\infty), \,\,0<\alpha<1,$$ with a non-random initial condition $u(x,0)=u_0(x),\,x\in\mathbb{R}$ assumed to be non-negative and bounded, $T_{\alpha,t}^a$ is a conformable time - fractional derivative, $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is globally Lipschitz continuous, $\dot{W}_t$ a generalized derivative of Wiener process and $\lambda>0$ is the noise level. Given some precise and suitable conditions on the non-random initial function, we study the asymptotic behaviour of the solution with respect to the time parameter $t$ and the noise level parameter $\lambda$. We also show that when the non-linear term $\sigma$ grows faster than linear, the energy of the solution blows-up at finite time for all $\alpha\in (0,1)$., Comment: 15 pages, submitted for publication, some crutial corrections to the previous version is made

The class of $\eta$-quasiconvex functions was introduced in 2016. Here we establish novel inequalities of Ostrowski type for functions whose second derivative, in absolute value raised to the power $q\geq 1$, is $\eta$-quasiconvex. Several interesting inequalities are deduced as special cases. Furthermore, we apply our results to the arithmetic, geometric, Harmonic, logarithmic, generalized log and identric means, getting new relations amongst them., Comment: This is a preprint of a paper whose final and definite form is accepted 19-Sept-2018 as a book chapter at Springer New York, on the topic of 'Frontiers in Functional Equations and Analytic Inequalities', Edited by G. Anastassiou and J. Rassias

We establish new integral inequalities of Hermite-Hadamard type for the recent class of $\eta$-convex functions. This is done via generalized $(k,r)$-Riemann-Liouville fractional integral operators. Our results generalize some known theorems in the literature. By choosing different values for the parameters $k$ and $r$, one obtains interesting new results., Comment: This is a preprint of a paper whose final and definite form is a Springer chapter in the Book 'Advances in Mathematical Inequalities and Applications', published under the Birkhauser series 'Trends in Mathematics', ISSN: 2297-0215 [see http://www.springer.com/series/4961]. Submitted 02-Jan-2018; Revised 10-Jan-2018; Accepted 13-Feb-2018

The aim of this paper is to generalize the Hermite--Hadamard inequality for functions convex on the coordinates. Our composite result generalizes the result of Dragomir in \cite{Drag}. Many other interesting inequalities can be derived from our results by choosing different values of $n\in\mathbb{N}.$ Furthermore, we add to the literature a new result for positive functions convex on the coordinates., Comment: 9 pages. This is a preprint of a paper whose final and definite form is published open access in Applied Mathematics E-Notes

We prove generalized weighted Ostrowski and Ostrowski--Gr\"uss type inequalities on time scales via a parameter function. In particular, our result extends a result of Dragomir and Barnett. Furthermore, we apply our results to the continuous, discrete, and quantum cases, to obtain some interesting new inequalities., Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Mathematical Inequalities' (JMI), ISSN: 1846-579X (print), 1848-9575 (online), available at [http://jmi.ele-math.com]. Paper Submitted 23-Dec-2016; Revised 21-June-2017; Accepted 27-June-2017

In this paper, we present a time scale version of the Hermite-Hadamard inequality for functions convex on the coordinates via the diamond-$\alpha$ calculus. Our results are new and they generalize and extend a result due to Dragomir., Comment: 13 pages

In this work, we obtain some new generalized weighted trapezoid and Gr\"uss type inequalities on time scales for parameter functions. Our results give a broader generalization of the results due to Pachpatte in \cite{Pach}. In addition, the continuous and discrete cases are also considered from which, other results are obtained.

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$. Rivlin \cite{Rivlin} proved that if $p(z)\neq 0$ in the unit disk, then for $0

We develop the Benkhettou-Hassani-Torres fractional (noninteger order) calculus on time scales by proving two chain rules for the $\alpha$-fractional derivative and five inequalities for the $\alpha$-fractional integral. The results coincide with well-known classical results when the operators are of (integer) order $\alpha = 1$ and the time scale coincides with the set of real numbers., Comment: This is a preprint of a paper whose final and definite form is published in 'Arabian Journal of Mathematics', ISSN 2193-5343 (Print) 2193-5351 (Online). Paper submitted 25/Feb/2016; revised 24/Sept/2016; accepted 28/Nov/2016

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$ having no zeros in the unit disk. ~Then it is well known that for $R\geq 1,$ $\displaystyle{\max_{|z|=R}|p(z)|}\leq \Big(\dfrac{R^n+1}{2}\Big)\displaystyle{\max_{|z|=1}|p(z)|}.$ In this paper, we consider polynomials with gaps, having all its zeros on the circle $S(0, K):=\{z: |z|=K\}, ~0

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n,$ where the coefficients $a_j,$ $j \in \{0,1,2,\cdots n\},$ are real numbers. We impose some restriction on the coefficients and then prove some extensions and generalizations of the Enestr\"{o}m-Kakeya Theorem., Comment: This is a preprint of a paper whose final and definite form is published open access in the Acta et Commentationes Universitatis Tartuensis de Mathematica (ACUTM)

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n,$ where the coefficients $a_j,$ $j \in \{0,1,2,\cdots n\},$ may be complex. We impose some restriction on the coefficients of the real part of the given polynomial and then estimate the maximum number of zeros such polynomial can possibly have in a specified disk., Comment: This is a preprint of a paper whose final and definite form is published open access in the International Journal of Pure and Applied Mathematics