An extension of an inequality for ratios of gamma functions

In this paper, we prove that for $x+y>0$ and $y+1>0$ the inequality {equation*} \frac{[\Gamma(x+y+1)/\Gamma(y+1)]^{1/x}}{[\Gamma(x+y+2)/\Gamma(y+1)]^{1/(x+1)}} 1$and reversed if$x<1$and that the power$\frac12$is the best possible, where$\Gamma(x)$ is the Euler gamma function. This extends the result in [Y. Yu, \textit{An inequality for ratios of gamma functions}, J. Math. Anal. Appl. \textbf{352} (2009), no.~2, 967\nobreakdash--970.] and resolves an open problem posed in [B.-N. Guo and F. Qi, \emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2, 239\nobreakdash--247.].Comment: 8 page

Path Integral Method for Pricing Proportional Step Double-Barrier Option with Time Dependent Parameters

Path integral method in quantum mechanics provides a new thinking for barrier option pricing. For proportional double-barrier step (PDBS) options, the option price changing process is analogous to a particle moving in a finite symmetric square potential well. We have derived the pricing kernel of PDBS options with time dependent interest rate and volatility. Numerical results of option price as a function of underlying asset price are shown as well. Path integral method can be easily generalized to the pricing of PDBS options with curved boundaries.Comment: 18 pages, 3 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2209.1254

Path Integral Method for Barrier Option Pricing Under Vasicek Model

Path integral method in quantum theory provides a new thinking for time dependent option pricing. For barrier options, the option price changing process is similar to the infinite high barrier scattering problem in quantum mechanics; for double barrier options, the option price changing process is analogous to a particle moving in a infinite square potential well. Using path integral method, the expressions of pricing kernel and option price under Vasicek stochastic interest rate model could be derived. Numerical results of options price as functions of underlying prices are also shown.Comment: 13pages,3figure

QCD corrections to single slepton production at hadron colliders

We evaluate the cross section for single slepton production at hadron colliders in supersymmetric theories with R-parity violating interactions to the next-to-leading order in QCD. We obtain fully differential cross section by using the phase space slicing method. We also perform soft-gluon resummation to all order in $\alpha_s$ of leading logarithm to obtain a complete transverse momentum spectrum of the slepton. We find that the full transverse momentum spectrum is peaked at a few GeV, consistent with the early results for Drell-Yan production of lepton pairs. We also consider the contribution from gluon fusion via quark-triangle loop diagrams dominated by the $b$-quark loop. The cross section of this process is significantly smaller than that of the tree-level process induced by the initial $b\bar{b}$ annihilation.Comment: one new reference is adde