Let $ {\rm{Ai}} $ and $ {\rm{Bi}} $ denote the Airy functions. For a fixed $k>0$, we introduce the class ${\rm{Airy}}_k(\mathcal I)$ of $k$–Airy convex functions on an interval $\mathcal I \subset [0, \infty)$. 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{{\rm{Ai}}(kt)\, {\rm{Bi}}(kb)-{\rm{Ai}}(kb)\, {\rm{Bi}}(kt)}{{\rm{Ai}}(ka)\, {\rm{Bi}}(kb)-{\rm{Ai}}(kb)\, {\rm{Bi}}(ka)}\, f(a) \;+\; \frac{{\rm{Ai}}(ka)\, {\rm{Bi}}(kt)-{\rm{Ai}}(kt)\, {\rm{Bi}}(ka)}{{\rm{Ai}}(ka)\, {\rm{Bi}}(kb)-{\rm{Ai}}(kb)\, {\rm{Bi}}(ka)}\, f(b). $
We establish fundamental properties of ${\rm{Airy}}_k(\mathcal I)$. As an application, we derive lower bounds for eigenvalues in Airy-type boundary value problems.
Citation: Bessem Samet. On $ k $–Airy convexity and applications to eigenvalue problems[J]. Networks and Heterogeneous Media, 2026, 21(2): 476-495. doi: 10.3934/nhm.2026022
Let $ {\rm{Ai}} $ and $ {\rm{Bi}} $ denote the Airy functions. For a fixed $k>0$, we introduce the class ${\rm{Airy}}_k(\mathcal I)$ of $k$–Airy convex functions on an interval $\mathcal I \subset [0, \infty)$. 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{{\rm{Ai}}(kt)\, {\rm{Bi}}(kb)-{\rm{Ai}}(kb)\, {\rm{Bi}}(kt)}{{\rm{Ai}}(ka)\, {\rm{Bi}}(kb)-{\rm{Ai}}(kb)\, {\rm{Bi}}(ka)}\, f(a) \;+\; \frac{{\rm{Ai}}(ka)\, {\rm{Bi}}(kt)-{\rm{Ai}}(kt)\, {\rm{Bi}}(ka)}{{\rm{Ai}}(ka)\, {\rm{Bi}}(kb)-{\rm{Ai}}(kb)\, {\rm{Bi}}(ka)}\, f(b). $
We establish fundamental properties of ${\rm{Airy}}_k(\mathcal I)$. As an application, we derive lower bounds for eigenvalues in Airy-type boundary value problems.
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Bessem Samet. On $ k $–Airy convexity and applications to eigenvalue problems[J]. Networks and Heterogeneous Media, 2026, 21(2): 476-495. doi: 10.3934/nhm.2026022
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