2000 Mathematics Subject Classification. Primary 26A48, 26A51; Secondary 26B25, 26D07. Key words and phrases. **Logarithmic** **convexity**, monotonicity, one-parameter mean values. The first author was supported in part by the Research Grants Council of the Hong Kong SAR (Project No. HKU7040/03P), CHINA. The second author was supported in part by SF for the Prominent Youth of Henan Province (#0112000200), SF of Henan Innovation Talents at Universities, Doctor Fund of Henan University of Technology, CHINA.

In the recent papers [21, 23, 37, 38, 39, 44, 45, 46, 47, 50], among other things, some analytic properties, including the general expression and a generalization of the asymptotic expansion (1.2), the monotonicity, **logarithmic** **convexity**, (logarith- mically) complete monotonicity, minimality, Schur-**convexity**, product and determi- nantal inequalities, exponential representations, integral representations, a gener- ating function, connections with the Bessel polynomials and the Bell polynomials of the second kind, and identities, of the Catalan numbers C n , the Catalan function

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In this article, we give more generalized results than in Anwar et al. (2010) and Latif and Pečarić (2010) in new direction by using second-order divided difference. We investigate the exponential **convexity** and **logarithmic** **convexity** for majorization type results by using class of continuous functions in linear functionals. We also construct positive semi-definite matrices for majorization type results. We will vary on choice of a family of functions in order to construct different examples of exponentially convex functions and construct some means. We also prove the monotonic property. Mathematics Subject Classification (2000): 26A51; 39B62; 26D15; 26D20; 26D99.

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2000 Mathematics Subject Classification. 26D15, 26A48, 26A51, 33B10, 33B15, 44A10, 65R10. Key words and phrases. bound, inequality, ratio of two gamma functions, psi function, polygamma function, divided difference, monotonicity, **logarithmic** **convexity**, completely mono- tonic function, logarithmically completely monotonic function, refinement, extension, generaliza- tion, exponential function, transform.

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applied a **logarithmic** **convexity** technique to achieve continuous dependence on the supply terms and structural stability on the coupling term for the classi- cal linear theory of thermoelasticity. They did not require the elasticity tensor to be sign-deﬁnite. All they needed was that the elasticity coeﬃcients were symmetric.

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We use the Lagrange identity method and the **logarithmic** **convexity** to obtain uniqueness and exponential growth of solutions in the thermoelasticity of type III and thermoelasticity without energy dissipation. As this is not the ﬁrst contri- bution of this kind in this theory, it is worth remarking that the assumptions we use here are diﬀerent from those used in other previous contributions. We assume that the elasticity tensor is positive semideﬁnite, but we allow that the constitu- tive tensor of the entropy ﬂux vector (k ij ), which is a characteristic tensor in this theory, is not sign-deﬁnite. The Lagrange identity method is used to obtain uniqueness in the context of the thermoelasticity of type III. The fundamental key to obtain exponential growth in the thermoelasticity without energy dissipation is the use of a new functional. This functional is inspired in that it is used when the elasticity tensor is not sign-deﬁnite, but (k ij ) is positive deﬁnite.

maximum cardinality of a proper weak convex set of G , wcon ( G ) = max { S / S is a weak convex set of G and S ≠ V (G )} . These type of sets are already called isometric sets. We prefer to use the term weak convex sets since the discussions are related to the **convexity** and the results there in. Also the condition of **convexity** is relaxed and hence we use the word weak convex. If S is a weak convex set in a connected graph G , then the subgraph S induced by S is connected. Weak convex set S in G with

In Section , we present an integral version of some results recently proved in []. We deﬁne linear functionals constructed from the non-negative diﬀerence of the reﬁned in- equalities and give mean value theorems for the linear functionals. In Section , we give deﬁnitions and results that will be needed later. Further, we investigate the n-exponential **convexity** and log-**convexity** of the functions associated with the linear functionals and also deduce Lyapunov-type inequalities. We also prove the monotonicity property of the generalized Cauchy means obtained via these functionals. Finally, in Section , we give several examples of the families of functions for which the obtained results can be applied.

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In the first part of this work, a convex partition of a compact subset is constructed. Minimum- length surrounding curve and minimum-area surrounding surfaces for a compact set are con- structed too. In the second part, one writes the perimeter of an ellipse as the sum of an alternate series. On the other hand, we deduce related “sandwich” inequalities for the perimeter, involving Jensen’s inequality and **logarithmic** function respectively. We discuss the values of the ordinate of the gravity center of the upper semiellipse at the ends of the positive semiaxes, in terms of the scale ratio b a .

Abstract:- Good News to all the Global population and mathematical scholars! It is focused that the Tamil based Indians who already lived in Mars Planet in prehistoric time before modern human population started leaving on Earth Planet. When lived in Mars they were very expert in Astronomical Science and climate control techniques and effectively controlling the inter planet disorders. They were not only expert in astronomical science but also expert in advanced mathematics and formulated the characteristics of planets, solar system, shape of Cosmo universe etc in tiny dot code form. They observe that the expansion of Universe matters and Physical and Chemical Properties of observable universe have **logarithmic** behavior. It is speculated that they adopted the mathematical system called NATURAL LOGARITHM for formulating the behavior of Universe, with constants such as e, pie(). Further it is speculated that these natural universal constants were of rational numbers in Prehistoric time.

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The research reported here was performed whilst Elodie Vernet was a student at ENS Cachan and visiting ANU and NICTA, and was supported by the Australian Research Council and NICTA, which was funded by the Australian government through the ICT centre of excellence program. An earlier version of some of these results appeared in NIPS2011 and ICML2012. The work benefited from discussions with Jake Abernethy, Tim van Erven, Rafael Frongillo, and Dario Garc´ıa-Garc´ıa, and comments from the referees who we thank. Thanks also to Harish Gu- ruprasad for identifying a flaw in an earlier proof of quasi-**convexity** of proper losses.

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We begin with a review of classical results about rearrangements, majorization and doubly stochastic operators on the space of functions. These are the key notions needed in the infinite-dimensional formulation of Schur-Horn **convexity** theorem. Our main reference for the material of this section is [19]. We should mention that the definition of doubly stochastic operators and their main properties have been studied rigorously for the first time by J. Ryff [63, 64, 65]. We restrict our attention to finite measure spaces which are easier to work with in rearrangement theory and which are sufficient for our purposes.

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In this paper, we present some reﬁnements of the classical Hermite-Hadamard integral inequality for convex functions. Further, we give the concept of n-exponential **convexity** and log-**convexity** of the functions associated with the linear functionals deﬁned by these inequalities and prove monotonicity property of the generalized Cauchy means obtained via these functionals. Finally, we give several examples of the families of functions for which the results can be applied.

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Motivated by earlier research works [, –] and by the importance of the concepts of **convexity** and generalized **convexity**, we discuss a new class of sets on Riemannian man- ifolds and a new class of functions deﬁned on them, which are called geodesic strongly E-convex sets and geodesic strongly E-convex functions, and some of their properties are presented.

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[13] J. S´ andor, On an inequality of Alze, J. Math. Anal. Appl. 192 (1995), 1034–1035. [14] J. S´ andor, On an inequality of Alzer, II, O. Math. Mag. 11 (2003), no. 2, 554–555. [15] K. B. Stolarsky, Generalizations of the **logarithmic** mean, Math. Mag. 48 (1975), 87–92. [16] K. B. Stolarsky, The power and generalized **logarithmic** means, Amer. Math. Monthly, 87

In this paper we use the idea of **logarithmic** density to deﬁne the concept of **logarithmic** statistical convergence. We ﬁnd the relations of **logarithmic** statistical convergence with statistical convergence, statistical summability (H, 1) introduced by Móricz (Analysis 24:127-145, 2004) and [H, 1] q -summability. We also give subsequence

We also give an improved upper bound of the **logarithmic** mean on Theorem . above, only for the Frobenius norm. Namely, we can prove the following inequalities in a similar way to the proof of Theorem ., by the use of scalar inequalities which will be given in Lemma ..

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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.

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deﬁne the modulus of **convexity** of a quasi-Banach space. In the third section, we estab- lish a relationship between the generalized von Neumann-Jordan constant and the mod- ulus of **convexity**, the James constant and the modulus of **convexity**, the generalized von Neumann-Jordan constant and the James constant, and we give the equivalent formula of the generalized von Neumann-Jordan constant.

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