Inequalities satisfied by the Airy functions (Q1970004)

The author establishes some general logarithmic convexity-concavity properties of the Airy functions \(Ai(x)\) and \(Bi(x)\) on certain intervals \(I_A\) and \(I_B\). He uses this information to derive inequalities such as \(xAi^(x)\leq Ai^2(x)\), \(Ai(x)Ai(y)\leq Ai^2((x+y)/2)\), and \(Ai^\alpha(x)Ai(0)^{1-\alpha}\leq Ai(\alpha x)\), \(0\leq\alpha\leq 1\), for \(x\in I_A\), with analogous results for the \(Bi(x)\). Sharp upper and lower bounds are determined for \(|Ai(z)|\) with \(z\in\mathbb{C}\) in terms of its restriction to the real axis. The author also derives an infinite product representation that he uses to approximate \(\log Ai(x)\) as \(x\to\infty\) without relying on any of the ``big'' results usually employed.