On $k$–Airy convexity and applications to eigenvalue problems

Let $\mathrm{A}\mathrm{i}$ and $\mathrm{B}\mathrm{i}$ denote the Airy functions. For a fixed $k>0$, we introduce the class ${\mathrm{A}\mathrm{i}\mathrm{r}\mathrm{y}}_k(\mathcal{I})$ of $k$–Airy convex functions on an interval $\mathcal{I}\subset [0,\mathrm{\infty })$. A function $f:\mathcal{I}\to \mathbb{R}$ is called $k$–Airy convex on $\mathcal{I}$ if, for every subinterval $[a,b]\subset \mathcal{I}$ and all $a<t<b$,

$$f(t)\le \frac{\mathrm{A}\mathrm{i}(kt)\mathrm{B}\mathrm{i}(kb)-\mathrm{A}\mathrm{i}(kb)\mathrm{B}\mathrm{i}(kt)}{\mathrm{A}\mathrm{i}(ka)\mathrm{B}\mathrm{i}(kb)-\mathrm{A}\mathrm{i}(kb)\mathrm{B}\mathrm{i}(ka)}f(a)+\frac{\mathrm{A}\mathrm{i}(ka)\mathrm{B}\mathrm{i}(kt)-\mathrm{A}\mathrm{i}(kt)\mathrm{B}\mathrm{i}(ka)}{\mathrm{A}\mathrm{i}(ka)\mathrm{B}\mathrm{i}(kb)-\mathrm{A}\mathrm{i}(kb)\mathrm{B}\mathrm{i}(ka)}f(b).$$

We establish fundamental properties of ${\mathrm{A}\mathrm{i}\mathrm{r}\mathrm{y}}_k(\mathcal{I})$. As an application, we derive lower bounds for eigenvalues in Airy-type boundary value problems.