logarithmic convexity

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Logarithmic Convexity of the One-Parameter Mean Values

Logarithmic Convexity of the One-Parameter Mean Values

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.

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SOME PROPERTIES OF THE FUSS–CATALAN NUMBERS

SOME PROPERTIES OF THE FUSS–CATALAN NUMBERS

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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Exponential convexity for majorization

Exponential convexity for majorization

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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Bounds for the Ratio of Two Gamma Functions

Bounds for the Ratio of Two Gamma Functions

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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Continuous dependence of solutions in magneto elasticity theory

Continuous dependence of solutions in magneto elasticity theory

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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Some remarks on growth and uniqueness in thermoelasticity

Some remarks on growth and uniqueness in thermoelasticity

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.

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The Weak (Monophonic) Convexity Number of a Graph

The Weak (Monophonic) Convexity Number of a Graph

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

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On the refinements of the integral Jensen Steffensen inequality

On the refinements of the integral Jensen Steffensen inequality

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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On Convexity and Approximating the Perimeter of an Ellipse

On Convexity and Approximating the Perimeter of an Ellipse

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 .

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Mars Mathematics (New Discovery on Value of E, Pie ( ) Constants)

Mars Mathematics (New Discovery on Value of E, Pie ( ) Constants)

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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Composite Multiclass Losses

Composite Multiclass Losses

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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A Convexity Theorem For Symplectomorphism Groups

A Convexity Theorem For Symplectomorphism Groups

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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On the refinements of the Hermite Hadamard inequality

On the refinements of the Hermite Hadamard inequality

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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On geodesic strongly E convex sets and geodesic strongly E convex functions

On geodesic strongly E convex sets and geodesic strongly E convex functions

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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On Integral Version of Alzer's Inequality and Martins' Inequality

On Integral Version of Alzer's Inequality and Martins' Inequality

[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

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Logarithmic density and logarithmic statistical convergence

Logarithmic density and logarithmic statistical convergence

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

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Bounds of the logarithmic mean

Bounds of the logarithmic mean

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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Exponential convexity of Petrović and related functional

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.

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Relations between generalized von Neumann Jordan and James constants for quasi Banach spaces

Relations between generalized von Neumann Jordan and James constants for quasi Banach spaces

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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