Saad Ihsan Butt

New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity

The inequality of Popoviciu, which was improved by Vasić and Stanković (Math. Balk. 6:281-288, 1976), is generalized by using new identities involving new Green’s functions. New generalizations of an improved Popoviciu inequality are obtained by using generalized Montgomery identity along with new Green’s functions. As an application, we formulate the monotonicity of linear functionals constructed from the generalized identities, utilizing the recent theory of inequalities for n-convex functions at a point. New upper bounds of Grüss and Ostrowski type are also computed.

Exponential convexity of Petrović and related functional

We consider functionals due to the difference in Petrović and related inequalities and prove the log-convexity and exponential convexity of these functionals by using different families of functions. We construct positive semi-definite matrices generated by these functionals and give some related results. At the end, we give some examples.

Refinement of integral inequalities for monotone functions

In this paper, we give refinements of some inequalities for generalized monotone functions by using log-convexity of some functionals.

New bounds for Shannon, Relative and Mandelbrot entropies via Hermite interpolating polynomial

Abstract To procure inequalities for divergences between probability distributions, Jensen’s inequality is the key to success. Shannon, Relative and Zipf-Mandelbrot entropies have many applications in many applied sciences, such as, in information theory, biology and economics, etc. We consider discrete and continuous cyclic refinements of Jensen’s inequality and extend them from convex function to higher order convex function by means of different new Green functions by employing Hermite interpolating polynomial whose error term is approximated by Peano’s kernal. As an application of our obtained results, we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.

Role of key players in paradigm shifts of prostate cancer bone metastasis

The decreased bone mineral density and compromised bone strength predispose individuals to skeletal osteoporosis. Both prostate cancer and bone metastasis caused by cancer invasion have remained a great challenge to researchers. With the advancement in the fields of biochemistry and biomechanics, the molecular mechanisms that make prostate cancer metastasize to bone have recently been identified, and they provide new molecular targets for drug development. Many biochemical by-products have been identified to help in understanding the interaction between the bone and the tumor. Enhanced clinical management of patients with bone metastases was reported during the past decade; however, the anticipated risk and the response to the therapy are still challenging to assess. In this review, the key players that play a dominant role in secondary osteoporosis are addressed. An attempt is made to provide the readers with a clear understanding of the communication pathways between each of the cell types involved in this vicious cycle. Furthermore, the role of Wnts, sclerostin, RANKL, PTHrP, and their respective clinical studies are addressed in this study.

Generalization of cyclic refinements of Jensen inequality by montgomery identity and Green’s function

We consider discrete and continuous cyclic refinements of Jensen’s inequality and generalize them from convex function to higher order convex function by means of Lagrange Green’s function and Montgomery identity. We give application of our results by formulating the monotonicity of the linear functionals obtained from generalized identities utilizing the theory of inequalities for n-convex functions at a point. We compute Gruss and Ostrowski type bounds for generalized identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity log convexity and mean value theorems.

Labelings of type (1, 1, 1) for Klein bottle fullerenes

In this paper we deal with the problem of labeling the vertices, edges and faces of a Klein bottle fullerenes

Superadditivity, Monotonicity, and Exponential Convexity of the Petrović-Type Functionals

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