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Search Results (5)

Search Parameters:
Keywords = Čebyšev inequality

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15 pages, 300 KiB  
Article
Some Properties on Normalized Tails of Maclaurin Power Series Expansion of Exponential Function
by Zhi-Hua Bao, Ravi Prakash Agarwal, Feng Qi and Wei-Shih Du
Symmetry 2024, 16(8), 989; https://doi.org/10.3390/sym16080989 - 5 Aug 2024
Cited by 10 | Viewed by 1747
Abstract
In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of [...] Read more.
In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementary function involving the exponential function, (4) with the help of an integral representation for the tail of the power series expansion of the exponential function, and (5) on account of Čebyšev’s integral inequality, the authors (i) expand the logarithm of the normalized tail of the power series expansion of the exponential function into a power series whose coefficients are expressed in terms of specific Hessenberg determinants whose elements are quotients of combinatorial numbers, (ii) prove the logarithmic convexity of the normalized tail of the power series expansion of the exponential function, (iii) derive a new determinantal expression of the Bernoulli numbers, deduce a determinantal expression for Howard’s numbers, (iv) confirm the increasing monotonicity of a function related to the logarithm of the normalized tail of the power series expansion of the exponential function, (v) present an inequality among three power series whose coefficients are reciprocals of combinatorial numbers, and (vi) generalize the logarithmic convexity of an extensively applied function involving the exponential function. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)
19 pages, 358 KiB  
Article
Generalized Čebyšev and Grüss Type Results in Weighted Lebesgue Spaces
by Saad Ihsan Butt, Josip Pečarić and Sanja Tipurić-Spužević
Mathematics 2023, 11(7), 1756; https://doi.org/10.3390/math11071756 - 6 Apr 2023
Cited by 1 | Viewed by 1172
Abstract
The classical Grüss and related inequalities have spurred a range of improvements, refinements, generalizations, and extensions. In the present article, we provide generalizations of Sokolov’s inequality in weighted Lebesgue LωΩ,A,μ spaces by employing the weighted Sonin’s identity [...] Read more.
The classical Grüss and related inequalities have spurred a range of improvements, refinements, generalizations, and extensions. In the present article, we provide generalizations of Sokolov’s inequality in weighted Lebesgue LωΩ,A,μ spaces by employing the weighted Sonin’s identity and Čebyšev functional. As a result, we provide a generalized Grüss inequality in which the bounding constants are improved with bounding functions in LωpΩ,A,μ spaces. As an application, we provide several new bounds for Jensen–Grüss type differences. Full article
14 pages, 299 KiB  
Article
Integral Results Related to Similarly Separable Vectors in Separable Hilbert Spaces
by Ravi P. Agarwal, Asif R. Khan and Sumayyah Saadi
Foundations 2022, 2(3), 813-826; https://doi.org/10.3390/foundations2030055 - 19 Sep 2022
Viewed by 1694
Abstract
In this work, we use similarly separable vectors in separable Hilbert spaces to provide generalized integral results related to majorization, Niezgoda, and Ćebysév type inequalities. Next, we furnish some refinements of these inequalities. Theorems obtained in this work extend and improve several known [...] Read more.
In this work, we use similarly separable vectors in separable Hilbert spaces to provide generalized integral results related to majorization, Niezgoda, and Ćebysév type inequalities. Next, we furnish some refinements of these inequalities. Theorems obtained in this work extend and improve several known results in the literature. An important aspect of our work is that these inequalities are directly related to Arithmetic, Geometric, Harmonic, and Power means. These means have played an important role in many branches of arts and sciences since the last 2600 years. Full article
(This article belongs to the Section Mathematical Sciences)
15 pages, 297 KiB  
Article
Some Bounds for the Complex Čebyšev Functional of Functions of Bounded Variation
by Silvestru Sever Dragomir
Symmetry 2021, 13(6), 990; https://doi.org/10.3390/sym13060990 - 2 Jun 2021
Viewed by 1872
Abstract
In this paper, we provide several bounds for the modulus of the complex Čebyšev functional. Applications to the trapezoid and mid-point inequalities, that are symmetric inequalities, are also provided. Full article
(This article belongs to the Special Issue Functional Equations and Analytic Inequalities)
8 pages, 268 KiB  
Article
Some Inequalities of Čebyšev Type for Conformable k-Fractional Integral Operators
by Feng Qi, Gauhar Rahman, Sardar Muhammad Hussain, Wei-Shih Du and Kottakkaran Sooppy Nisar
Symmetry 2018, 10(11), 614; https://doi.org/10.3390/sym10110614 - 8 Nov 2018
Cited by 41 | Viewed by 5720
Abstract
In the article, the authors present several inequalities of the Čebyšev type for conformable k-fractional integral operators. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
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