Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1705.06167

Bounds for the gamma function

Necdet Batir
View original source ↗Read source document ↗Export metadata ↓

Abstract

We improve the upper bound of the following inequalities for the gamma function $Γ$ due to H. Alzer and the author. \begin{equation*} \exp\left(-\frac{1}{2}ψ(x+1/3)\right)<\frac{Γ(x)}{x^xe^{-x}\sqrt{2π}}<\exp\left(-\frac{1}{2}ψ(x)\right). \end{equation*} We also prove the following new inequalities: For $x\geq1$ \[ \sqrt{2π}x^xe^{-x}\left(x^2+\frac{x}{3}+a_*\right)^{\frac{1}{4}}<Γ(x+1)<\sqrt{2π}x^xe^{-x}\left(x^2+\frac{x}{3}+a^*\right)^{\frac{1}{4}} \] with the best possible constants $a_*=\frac{e^4}{4π^2}-\frac{4}{3}=0.049653963176...$, and $a^*=1/18=0.055555...$, and for $x\geq0$ \begin{equation*} \exp\left[xψ\left(\frac{x}{\log (x+1)}\right)\right]\leqΓ(x+1)\leq\exp\left[xψ\left(\frac{x}{2}+1\right)\right], \end{equation*} where $ψ$ is the digamma function.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Necdet Batir. 2017-05-17. Bounds for the gamma function. https://arxiv.org/abs/1705.06167

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Let $p$ be an odd prime. We prove the extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every nonzero-radius sphere $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq \, 34/11$, uniformly in $p$ and $j$. This improves the Stein--Tomas exponent $10/3$ established by Iosevich and Koh (2008). We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture implies the spherical extension estimates $R_{S_j}^*(2\to r)\lesssim_r 1$ for every $r>3$, and yields almost-every-pin distance estimates at the conjectured Erdős--Falconer exponent in four dimensions, up to an arbitrarily small power loss in the set-size hypothesis. Using the same method, we improve the bounds supplied by Fourier decay and Plancherel at intermediate support scales and derive new almost-every-pin distance estimates in $\mathbb{F}_p^4$.

math.CA↗
arXiv2026

Dimension-free estimates for discrete maximal functions over cubes in $\mathbb Z^d$

In this short note, we establish dimension-free $\ell^p(\mathbb Z^d)$ bounds, for all $p\in(1,\infty]$, for the discrete Hardy--Littlewood maximal functions associated with cubes in $\mathbb Z^d$, answering a question that had been open for a while. The key idea is to prove dimension-free bounds for the $\ell^p(\mathbb Z^d)$ norms of the differences of the corresponding averages. This follows from an ad hoc interpretation of the associated discrete multipliers as a special continuous family of multipliers to which basic fractional integration and complex interpolation can be applied. The same method also yields an elementary proof of Bourgain's dimension-free $L^p(\mathbb R^d)$ bounds for the Hardy--Littlewood maximal function associated with cubes in $\mathbb R^d$.

math.CA↗
arXiv2026

Establishing the Polynomial Wolff Axioms for $δ$-Separated $δ$-Tubes With #o-minimality

We establish the full version of a conjecture of Guth and Zahl, giving a lower bound for the volume of a semialgebraic set that has a large intersection with a collection of $δ$-separated $δ$-tubes. Our proof uses o-minimal methods to simplify the proof of Katz and Rogers, who proved the conjecture up to a small factor. We also establish that the constants depend polynomially on the complexity of the semialgebraic set, and more generally in the #o-minimal setting.

math.CA↗