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Inequalities for generalized hypergeometric functions
AbstractIt is shown that some well-known Padé approximations for a particular form of the Gaussian hypergeometric function and two of its confluent forms give upper and lower bounds for these functions under suitable restrictions on the parameters and variable. With the aid of the beta and Laplace transforms, two-sided inequalities are derived for the generalized hypergeometric function pFq, p = q or p = q + 1, and for a particular form of Meijer's G-function. Several examples are developed. These include upper and lower bounds for certain elementary functions, complete elliptic integrals, the incomplete gamma function, modified Bessel functions, and parabolic cylinder functions
NEW DOCTORAL DEGREES Integral expressions for series of functions of hypergeometric and Bessel types
This thesis presents some new results on integral expressions for series of functions of hypergeometric and Bessel types. Also there are derived two--sided inequalities of some hypergeometric functions, which are related with their integral representations.
In the first part of the thesis are defined some special functions, mathematical methods, and results which we use in prooving our own. Some of them are Gamma function, Gauss hypergeometric function and generalized hypergeometric function . There are also Fox-Wright generalized hypergeometric function and the Struve function .\
Bessel differential equation is also described, and that is one of the crucial mathematical tools that we use.\
Mathieu -- and Dirichlet series are defined too, because they are useful for deriving most of integral representations. In that purpose, we also use condensed form of Euler--Maclaurin summation formula and fractional analysis, which are described in the introduction.
In the middle part of the thesis, i.e. in Chapter 3, 4 and 5 we work on integral representations of functional series with members containing Bessel functions of the first kind, which are divided into three types: Neumann series, which are discussed in Chapter 3, Kapteyn series, which are described in Chapter 4, and Schl"omilch series, which are observed in Chapter 5.
In the last chapter of this thesis, we obtain a functional series of hypergeometric types. There, we also derive an integral representations of hypergeometric functions, such as extended general Hurwitz--Lerch Zeta function and extended Hurwitz--Lerch Zeta function, and also the two-sided inequalities for the mentioned functions.
\At the end of this chapter, new incomplete generalized Hurwitz--Lerch Zeta functions and incomplete generalized Gamma functions are defined, and we also investigate their important properties
A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications
A class of functions involving the divided differences of the psi function
and the polygamma functions and originating from Kershaw's double inequality
are proved to be completely monotonic. As applications of these results, the
monotonicity and convexity of a function involving ratio of two gamma functions
and originating from establishment of the best upper and lower bounds in
Kershaw's double inequality are derived, two sharp double inequalities
involving ratios of double factorials are recovered, the probability integral
or error function is estimated, a double inequality for ratio of the volumes of
the unit balls in and respectively is
deduced, and a symmetrical upper and lower bounds for the gamma function in
terms of the psi function is generalized.Comment: 11 page
New Upper Bounds in the Second Kershaw's Double Inequality and its Generalizations
In the paper, new upper bounds in the second Kershaw’s double
inequality and its generalizations involving the gamma, psi and polygamma
functions are established, some known results are refined
On the expansion of the Kummer function in terms of incomplete Gamma functions
The expansion of Kummer's hypergeometric function as a series of incomplete
Gamma functions is discussed, for real values of the parameters and of the
variable. The error performed approximating the Kummer function with a finite
sum of Gammas is evaluated analytically. Bounds for it are derived, both
pointwisely and uniformly in the variable; these characterize the convergence
rate of the series, both pointwisely and in appropriate sup norms. The same
analysis shows that finite sums of very few Gammas are sufficiently close to
the Kummer function. The combination of these results with the known
approximation methods for the incomplete Gammas allows to construct upper and
lower approximants for the Kummer function using only exponentials, real powers
and rational functions. Illustrative examples are provided.Comment: 21 pages, 6 figures. To appear in "Archives of Inequalities and
Applications
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