这是1992年的预印本，有一些更新。研究了指数函数泰勒级数的截面。G.Hardy，Kesava Menon，W. Gautschi，H.Alzer and others.本预印本的主要目的是研究具有最佳常数的主要不等式及其多重推广的新证明。一些公式化（其中一些最近被证明，见2016年的评论）。
英文摘要：
This is a preprint of 1992 with some updates. We study sections of the exponential function Taylor series. Interesting inequalities for these sections were considered by G.Hardy, Kesava Menon, W. Gautschi, H.Alzer and others. The main aim of this preprint is to investigate new proofs for the main inequality with best constants and its multiple generalizations. Some conjectures are formulated (and some of them were proved recently, see comments of 2016).
We obtain a remainder estimate for the truncated Taylor expansion for differential equations driven by weakly geometric $Π$-rough paths for $Π=\left( p_{1},\cdots ,p_{k}\right) $, $p_{i}\geq 1$. When there exists $ p\geq 1$ such that $p_{i}=pk_{i}^{-1}\ $for some $k_{i}\in \left\{ 1,\dots , \left[ p\right] \right\} $, we obtain a refined Taylor remainder estimate that contains a factorial decay component. The remainder estimates are in the right order as they are comparable to the next term in the Taylor expansion.
Motivated by several conjectures posed in the paper " Completely monotonic degrees for a difference between the logarithmic and psi functions",we confirm in this work some conjectures on completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function and the digamma function and we give two bounded for this degrees.
This paper is devoted to a new first order Taylor-like formula where the corresponding remainder is strongly reduced in comparison with the usual one which appears in the classical Taylor's formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at n+1 equally-spaced points between the two points where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange P1- interpolation error est…
In this paper, we deal with the convolution series that are a far reaching generalization of the conventional power series and the power series with the fractional exponents including the Mittag-Leffler type functions. Special attention is given to the most interesting case of the convolution series generated by the Sonine kernels. In this paper, we first formulate and prove the second fundamental theorem for the general fractional integrals and the $n$-fold general sequential fractional derivatives of both the Riemann-Liouville and the Caputo types. These results are then employed for derivat…
Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we proved that $$ H(q) := T(q)- \frac{\log(1-q)}{\log(q)} $$ is strictly increasing in $ (0,1) $ while $$ F(q) := \frac{1-q}{q} \,H(q) $$ is strictly decreasing in $ (0,1) $. These results are then applied to obtain various inequalities, one of which states that the double-inequality $$ α\,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)} < T(q)< β\,\frac{q}{1-q}+\frac{\log(1-q)}…
We show that on $\mathbb S^1(1/\sqrt{d-2})\times\mathbb S^{d-1}(1)$ the conformally invariant Sobolev inequality holds with a remainder term that is the fourth power of the distance to the optimizers. The fourth power is best possible. This is in contrast to the more usual vanishing to second order and is motivated by work of Engelstein, Neumayer and Spolaor. A similar phenomenon arises for subcritical Sobolev inequalities on $\mathbb S^d$. Our proof proceeds by an iterated Bianchi-Egnell strategy.
If a smooth function of one variable has maximum one on the unit interval, and has there $d$ zeroes, then its $(d+1)$-st derivative must be "big". This is one of the simplest examples of what we call "smooth rigidity": certain geometric properties of zero sets of smooth functions $f$ imply explicit lower bounds on the high-order derivatives of $f$. In dimensions greater than one, the powerful one-dimension tools, like Lagrange's remainder formula, and divided finite differences, are not directly applicable. Still, the result above implies, via line sections, rather strong restrictions on zeroe…
Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz's norm, the space…
