scholarly journals Lower Bounds for the Counting Function of Integral Operators

2012 ◽  
pp. 343-354
Author(s):  
Y. Safarov
Keyword(s):  
Lower Bounds ◽  
Counting Function ◽  
Integral Operators

scholarly journals Upper and lower bounds of integral operator defined by the fractional hypergeometric function

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Rabha W. Ibrahim ◽  
Muhammad Zaini Ahmad ◽  
Hiba F. Al-Janaby
Keyword(s):  
Integral Operator ◽  
Hypergeometric Function ◽  
Lower Bounds ◽  
Analytic Functions ◽  
Hypergeometric Functions ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Holomorphic Functions ◽  
Upper And Lower Bounds ◽  
Distortion Theorems

AbstractIn this article, we impose some studies with applications for generalized integral operators for normalized holomorphic functions. By using the further extension of the extended Gauss hypergeometric functions, new subclasses of analytic functions containing extended Noor integral operator are introduced. Some characteristics of these functions are imposed, involving coefficient bounds and distortion theorems. Further, sufficient conditions for subordination and superordination are illustrated.

scholarly journals Study of inequalities for unified integral operators of generalized convex functions

2021 ◽  
Vol 5 (1) ◽  
pp. 80-93
Author(s):  
G. Farid ◽  
◽  
K. Mahreen ◽  
Yu-Ming Chu ◽  
◽  
...  
Keyword(s):  
Lower Bounds ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Generalized Convex Functions ◽  
Fractional Integral Operators

The aim of this paper is to study unified integral operators for generalized convex functions namely \((\alpha,h-m)\)-convex functions. We obtained upper as well as lower bounds of these integral operators in diverse forms. The results simultaneously hold for many kinds of well known fractional integral operators and for various kinds of convex functions.

scholarly journals On the Spectral Asymptotics of Operators on Manifolds with Ends

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Sandro Coriasco ◽  
Lidia Maniccia
Keyword(s):  
Asymptotic Behaviour ◽  
Pseudodifferential Operators ◽  
Counting Function ◽  
Integral Operators ◽  
Fourier Integral ◽  
Spectral Asymptotics ◽  
Fourier Integral Operators ◽  
Noncompact Manifolds ◽  
Adjoint Operators

We deal with the asymptotic behaviour, forλ→+∞, of the counting functionNP(λ)of certain positive self-adjoint operatorsPwith double order(m,μ),m,μ>0,  m≠μ, defined on a manifold with endsM. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier integral operators associated with weighted symbols globally defined onℝn. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae forNP(λ)and show how their behaviour depends on the ratiom/μand the dimension ofM.

scholarly journals Derivation of bounds of several kinds of operators via $(s,m)$-convexity

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Young Chel Kwun ◽  
Ghulam Farid ◽  
Shin Min Kang ◽  
Babar Khan Bangash ◽  
Saleem Ullah
Keyword(s):  
Lower Bounds ◽  
Convex Functions ◽  
Integral Operators ◽  
Upper And Lower Bounds ◽  
Hadamard Inequality

AbstractThe objective of this paper is to derive the bounds of fractional and conformable integral operators for $(s,m)$(s,m)-convex functions in a unified form. Further, the upper and lower bounds of these operators are obtained in the form of a Hadamard inequality, and their various fractional versions are presented. Some connections with already known results are obtained.

scholarly journals Derivation of Bounds of an Integral Operator via Exponentially Convex Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Hong Ye ◽  
Ghulam Farid ◽  
Babar Khan Bangash ◽  
Lulu Cai
Keyword(s):  
Integral Operator ◽  
Lower Bounds ◽  
Differentiable Function ◽  
Convex Functions ◽  
Integral Operators ◽  
Compact Form ◽  
Upper And Lower Bounds ◽  
Hadamard Inequality ◽  
Absolute Value ◽  
Exponentially Convex Functions

In this paper, bounds of fractional and conformable integral operators are established in a compact form. By using exponentially convex functions, certain bounds of these operators are derived and further used to prove their boundedness and continuity. A modulus inequality is established for a differentiable function whose derivative in absolute value is exponentially convex. Upper and lower bounds of these operators are obtained in the form of a Hadamard inequality. Some particular cases of main results are also studied.

scholarly journals ON THE BITS COUNTING FUNCTION OF REAL NUMBERS

2008 ◽  
Vol 85 (1) ◽  
pp. 95-111 ◽  
Author(s):  
TANGUY RIVOAL
Keyword(s):  
Lower Bounds ◽  
Irrational Number ◽  
Counting Function ◽  
Fibonacci Sequence ◽  
Difficult Problem ◽  
Transcendental Number ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Binary Expansion ◽  
Multiplicative Constant

AbstractLet Bn(x) denote the number of 1’s occurring in the binary expansion of an irrational number x>0. A difficult problem is to provide nontrivial lower bounds for Bn(x) for interesting numbers such as $\sqrt {2}$, e or π: their conjectural simple normality in base 2 is equivalent to Bn(x)∼n/2. In this article, amongst other things, we prove inequalities relating Bn(x+y), Bn(xy) and Bn(1/x) to Bn(x) and Bn(y) for any irrational numbers x,y>0, which we prove to be sharp up to a multiplicative constant. As a by-product, we provide an answer to a question raised by Bailey et al. (D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, ‘On the binary expansions of algebraic numbers’, J. Théor. Nombres Bordeaux16(3) (2004), 487–518) concerning the binary digits of the square of a series related to the Fibonacci sequence. We also obtain a slight refinement of the main theorem of the same article, which provides a nontrivial lower bound for Bn(α) for any real irrational algebraic number. We conclude the article with effective or conjectural lower bounds for Bn(x) when x is a transcendental number.

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