Displaying similar documents to “Bound on extended f -divergences for a variety of classes”

Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Sever S. Dragomir (2015)

Annales Mathematicae Silesianae

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Let [...] f(z)=∑n=0∞αnzn f ( z ) = n = 0 α n z n be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ ℂ, R > 0. For any x, y ∈ ℬ, a Banach algebra, with ‖x‖, ‖y‖ < R we show among others that [...] ‖f(y)−f(x)‖≤‖y−x‖∫01fa′(‖(1−t)x+ty‖)dt f ( y ) - f ( x ) y - x 0 1 f a ' ( ( 1 - t ) x + t y ) d t where [...] fa(z)=∑n=0∞|αn| zn f a ( z ) = n = 0 | α n | z n . Inequalities for the commutator such as [...] ‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖, f ( x ) f ( y ) - f ( y ) f ( x ) 2 f a ( M ) f a ' ( M ) y - x , if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided. ...

Ostrowski’s type inequalities for complex functions defined on unit circle with applications for unitary operators in Hilbert spaces

S.S. Dragomir (2015)

Archivum Mathematicum

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Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral a b f e i t d u t of continuous complex valued integrands f : 𝒞 0 , 1 defined on the complex unit circle 𝒞 0 , 1 and various subclasses of integrators u : a , b 0 , 2 π of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well.

Norm inequalities for the difference between weighted and integral means of operator differentiable functions

Silvestru Sever Dragomir (2020)

Archivum Mathematicum

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Let f be a continuous function on I and A , B 𝒮𝒜 I H , the convex set of selfadjoint operators with spectra in I . If A B and f , as an operator function, is Gateaux differentiable on [ A , B ] : = ( 1 - t ) A + t B t 0 , 1 , while p : 0 , 1 is Lebesgue integrable, then we have the inequalities 0 1 p τ f 1 - τ A + τ B d τ - 0 1 p τ d τ 0 1 f 1 - τ A + τ B d τ 0 1 τ ( 1 - τ ) | τ 1 p s d s 1 - τ - 0 τ p s d s τ | f 1 - τ A + τ B B - A d τ 1 4 0 1 | τ 1 p s d s 1 - τ - 0 τ p s d s τ | f 1 - τ A + τ B B - A d τ , where f is the Gateaux derivative of f .

Convex approximation of an inhomogeneous anisotropic functional

Giovanni Bellettini, Maurizio Paolini (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The numerical minimization of the functional F u = Ω ϕ x , ν u D u + Ω μ u d H n - 1 - Ω κ u d x , u B V Ω ; - 1 , 1 is addressed. The function ϕ is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that F can be equivalently minimized on the convex set B V Ω ; - 1 , 1 and then regularized with a sequence F ϵ u ϵ , of stricdy convex functionals defined on B V Ω ; - 1 , 1 . Then both F and F ϵ , can be discretized by continuous linear finite elements. The convexity property of the functionals on B V Ω ; - 1 , 1 is useful in the numerical...

Some distribution results on generalized ballot problems

Jagdish Saran, Kanwar Sen (1985)

Aplikace matematiky

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Suppose that in a ballot candidate A scores a votes and candidate B scores b votes and that all possible a + b a voting sequences are equally probable. Denote by α r and by β r the number of votes registered for A and for B , respectively, among the first r votes recorded, r = 1 , , a + b . The purpose of this paper is to derive, for a b - c , the probability distributions of the random variables defined as the number of subscripts r = 1 , , a + b for which (i) α r = β r - c , (ii) α r = β r - c but α r - 1 = β r - 1 - c ± 1 , (iii) α r = β r - c but α r - 1 = β r - 1 - c ± 1 and α r + 1 = β r + 1 - c ± 1 , where c = 0 , ± 1 , ± 2 , .

Improved upper bounds for nearly antipodal chromatic number of paths

Yu-Fa Shen, Guo-Ping Zheng, Wen-Jie HeK (2007)

Discussiones Mathematicae Graph Theory

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For paths Pₙ, G. Chartrand, L. Nebeský and P. Zhang showed that a c ' ( P ) n - 2 2 + 2 for every positive integer n, where ac’(Pₙ) denotes the nearly antipodal chromatic number of Pₙ. In this paper we show that a c ' ( P ) n - 2 2 - n / 2 - 10 / n + 7 if n is even positive integer and n ≥ 10, and a c ' ( P ) n - 2 2 - ( n - 1 ) / 2 - 13 / n + 8 if n is odd positive integer and n ≥ 13. For all even positive integers n ≥ 10 and all odd positive integers n ≥ 13, these results improve the upper bounds for nearly antipodal chromatic number of Pₙ.

A note on a property of the Gini coefficient

Marian Genčev (2019)

Communications in Mathematics

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The scope of this note is a self-contained presentation of a mathematical method that enables us to give an absolute upper bound for the difference of the Gini coefficients G ( σ 1 , , σ n ) - G ( γ 1 , , γ n ) , where ( γ 1 , , γ n ) represents the vector of the gross wages and ( σ 1 , , σ n ) represents the vector of the corresponding super-gross wages that is used in the Czech Republic for calculating the net wage. Since (as of June 2019) σ i = 100 · 1 . 34 γ i / 100 , the study of the above difference seems to be somewhat inaccessible for many economists. However, our estimate...

Inequalities for the arithmetical functions of Euler and Dedekind

Horst Alzer, Man Kam Kwong (2020)

Czechoslovak Mathematical Journal

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For positive integers n , Euler’s phi function and Dedekind’s psi function are given by φ ( n ) = n p n p prime 1 - 1 p and ψ ( n ) = n p n p prime 1 + 1 p , respectively. We prove that for all n 2 we have 1 - 1 n n - 1 1 + 1 n n + 1 φ ( n ) n φ ( n ) ψ ( n ) n ψ ( n ) and φ ( n ) n ψ ( n ) ψ ( n ) n φ ( n ) 1 - 1 n n + 1 1 + 1 n n - 1 . The sign of equality holds if and only if n is a prime. The first inequality refines results due to Atanassov (2011) and Kannan & Srikanth (2013).