In this paper we study the monotonicity properties of some functions involving the Mills' ratio
of the standard normal law. From these we deduce some new functional inequalities …

Consider the Mills ratio f (x)=(1− Φ (x))/ϕ (x), x≥ 0, where ϕ is the density function of the
standard Gaussian law and Φ its cumulative distribution. We introduce a general procedure …

We review various inequalities for Mills' ratio (1-\Phi)/\phi, where\phi and\Phi denote the
standard Gaussian density and distribution function, respectively. Elementary considerations …

The Mills ratio of the standard Gaussian law plays an important role in probability and
statistics since its reciprocal is the failure rate function. In the paper, we verify the increasing …

Let $${\rm erf}(x)=\frac {2}{\sqrt {\pi}}\int_0^ xe^{-t^{2}} dt $$ be the error function. We prove
that the following inequalities are valid for all positive real numbers x, y with $$ x\leq y …

We show several variants of concentration inequalities on the sphere stated as subgaussian
estimates with optimal constants. For a Lipschitz function, we give one-sided and two-sided …

In 1998, we have given in ([14] Intissar, A., Analyse de Scattering d'un op\'erateur cubique
de Heun dans l'espace de Bargmann, Comm. Math. Phys. 199 (1998) 243-256) the …

We provide an alternative proof of monotonicity of normalizing volatility transforms (NVTs)
due to Fukasawa (2012), and then obtain a general formula for volatility surface for which …

In this paper, we present some new proofs for asymptotic series of Mills' ratio. Given these
new inequalities, the upper bound of the error of Mills' ratio can be estimated more exactly …

Although there is an extensive literature on the upper bound for cumulative standard normal
distribution, there are relatively not sharp for all values of the interested argument x. The aim …