Lower Bounds for the Counting Function of an Integral Operator

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.

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Lower bounds for the generalized counting function

The Maz’ya Anniversary Collection ◽

10.1007/978-3-0348-8672-7_16 ◽

1999 ◽

pp. 275-293 ◽

Cited By ~ 1

Author(s):

Yu. Safarov

Keyword(s):

Lower Bounds ◽

Counting Function

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Derivation of Bounds of an Integral Operator via Exponentially Convex Functions

Journal of Mathematics ◽

10.1155/2020/2456463 ◽

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.

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ON THE BITS COUNTING FUNCTION OF REAL NUMBERS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788708000591 ◽

2008 ◽

Vol 85 (1) ◽

pp. 95-111 ◽

Cited By ~ 8

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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