On univalence of certain infinite products

Chernoff’s density is log-concave

“…This is also Lemma 1 of Salmassi [41]. Our statement of Proposition 2.1 corrects the constants c 1 and c 2 given by Merkes and Salmassi [34]. Figure 1 shows Ai(z) (black) and m term approximations to Ai(z) based on Proposition 2.1 with m = 25 (green), 125 (magenta), and 500 (blue).…”

“…1 is given by Merkes and Salmassi [34]; see their Lemma 1, page 211. This is also Lemma 1 of Salmassi [41].…”

“…Theorem 2.2 (Schoenberg, 1951). A necessary and sufficient condition for a (density) function g(x), −∞ < x < ∞, to be a PF ∞ (density) function is that the reciprocal of its bilateral Laplace transform (i.e., Fourier) be an entire function of the form where [34]; see their Lemma 1, page 211. This is also Lemma 1 of Salmassi [41].…”

Inequalities Satisfied by the Airy Functions

“…Below we give two proofs based on the infinite product representation of Ai(x). In [5] we used the latter to obtain the radius of univalence of Ai(z). The following lemma summarizes all that we need.…”

MHD α2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator

“…. over square wells can probably be achieved by representing the characteristic determinant ∆(E) in (54) via Hadamar product representation of the Airy functions [53] as a spectral determinant of Bethe-ansatz type [16,29,54] and studying it by similar cocycle functional equations as in Ref. [55].…”