scholarly journals Relative Growth of Series in Systems of Functions and Laplace—Stieltjes-Type Integrals

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 43
Author(s):  
Myroslav Sheremeta
Keyword(s):  
Asymptotic Behavior ◽  
Transcendental Function ◽  
Entire Transcendental Function ◽  
Relative Growth ◽  
Type Integral ◽  
Stieltjes Type

For a regularly converging-in-C series A(z)=∑n=1∞anf(λnz), where f is an entire transcendental function, the asymptotic behavior of the function Mf−1(MA(r)), where Mf(r)=max{|f(z)|:|z|=r}, is investigated. It is proven that, under certain conditions on the functions f, α, and the coefficients an, the equality limr→+∞α(Mf−1(MA(r)))α(r)=1 is correct. A similar result is obtained for the Laplace–Stiltjes-type integral I(r)=∫0∞a(x)f(rx)dF(x). Unresolved problems are formulated.

scholarly journals On the growth of series in system of functions and Laplace-Stieltjes integrals

2021 ◽  
Vol 55 (2) ◽  
pp. 124-131
Author(s):  
M.M. Sheremeta
Keyword(s):  
Transcendental Function ◽  
Entire Transcendental Function ◽  
Type Integral ◽  
System F ◽  
The Relationship ◽  
Stieltjes Type ◽  
Generalized Order

For a regularly convergent in ${\Bbb C}$ series $A(z)=\sum\nolimits_{n=1}^{\infty}a_nf(\lambda_nz)$ in the system ${f(\lambda_nz)}$, where$f(z)=\sum\nolimits_{k=0}^{\infty}f_kz^k$ is an entire transcendental function and $(\lambda_n)$is a sequence of positive numbers increasing to $+\infty$, it isinvestigated the relationship between the growth of functions $A$ and $f$ in terms of a generalized order. It is proved that if$a_n\ge 0$ for all $n\ge n_0$, $\ln \lambda_n=o\big(\beta^{-1}\big(c\alpha(\frac{1}{\ln \lambda_n}\ln \frac{1}{a_n})\big)\big)$ for each $c\in (0, +\infty)$ and $\ln n=O(\Gamma_f(\lambda_n))$ as $n\to\infty$ then $\displaystyle\varlimsup\limits_{r\to+\infty}\frac{\alpha(\ln M_A(r))}{\beta(\ln r)}=\varlimsup\limits_{r\to+\infty}\frac{\alpha(\ln M_f(r))}{\beta(\ln r)},$ where $M_f(r)=\max\{|f(z)|\colon |z|=r\}$, $\Gamma_f(r):=\frac{d\ln M_f(r)}{d\ln r}$ and positive continuous on $(x_0, +\infty)$ functions $\alpha$and $\beta$ are such that $\beta((1+o(1))x)=(1+o(1))\beta(x)$, $\alpha(c x)=(1+o(1))\alpha(x)$ and$\frac{d\beta^{-1}(c\alpha(x))}{d\ln x}=O(1)$ as $x\to+\infty$ for each $c\in(0, +\infty)$.\A similar result is obtained for the Laplace-Stieltjes type integral $I(r) = \int\limits_{0}^{\infty}a(x)f(rx) dF(x)$.

On a Choquet-Stieltjes type integral on intervals

2019 ◽  
Vol 69 (4) ◽  
pp. 801-814 ◽  
Author(s):  
Sorin G. Gal
Keyword(s):  
Lebesgue Measure ◽  
Choquet Integral ◽  
Stieltjes Integral ◽  
Line Integral ◽  
Type Integral ◽  
Choquet Integrals ◽  
Lebesgue Measures ◽  
Stieltjes Type

Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

scholarly journals Dynamics of exp (z)

1984 ◽  
Vol 4 (1) ◽  
pp. 35-52 ◽  
Author(s):  
Robert L. Devaney ◽  
Michal Krych
Keyword(s):  
Complex Plane ◽  
Symbolic Dynamics ◽  
Final Section ◽  
Julia Set ◽  
Dynamical Behaviour ◽  
Transcendental Function ◽  
Entire Transcendental Function ◽  
Orbit Structure ◽  
The Family ◽  
Entire Complex

AbstractWe describe the dynamical behaviour of the entire transcendental function exp(z). We use symbolic dynamics to describe the complicated orbit structure of this map whose Julia Set is the entire complex plane. Bifurcations occurring in the family c exp(z) are discussed in the final section.

Lyapunov Type Integral Inequalities for Certain Differential Equations

1997 ◽  
Vol 4 (2) ◽  
pp. 139-148
Author(s):  
B. G. Pachpatte
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Second Order ◽  
Integral Inequalities ◽  
Asymptotic Behavior Of Solutions ◽  
Second Order Differential Equations ◽  
Elementary Analysis ◽  
Type Integral ◽  
Zeros Of Solutions

Abstract In the present paper we establish Lyapunov type integral inequalities related to the zeros of solutions of certain second-order differential equations by using elementary analysis. We also present some immediate applications of our results to study the asymptotic behavior of solutions of the corresponding differential equations.

scholarly journals An arithmetic remark on entire periodic functions

1971 ◽  
Vol 5 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Kurt Mahler
Keyword(s):  
Transcendental Function ◽  
Periodic Functions ◽  
Entire Transcendental Function ◽  
Image Position ◽  
Rational Integral

For every positive number w, there exists an odd entire transcendental function.with rational integral coefficients ah such that f〈z+w〉 = f〈z〉.

Existence of an entire transcendental function of boundedl-index

1995 ◽  
Vol 57 (1) ◽  
pp. 88-90 ◽  
Author(s):  
A. A. Gol'dberg ◽  
M. N. Sheremeta
Keyword(s):  
Transcendental Function ◽  
Entire Transcendental Function
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