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Higher monotonicity properties of special functions: application on Bessel case $$| \nu | <$$. (English) Zbl 0721.34028
Consider an equation $$(1)\quad y''+q(t)y=0$$ and define $M_ i=\int^{t_{i+1}}_{t_ i}[p(t)]^{-\alpha}| y(t)|^{\lambda}dt,\quad i=0,1,...,$ where y(t) is an arbitrary nontrivial solution of (1) and $$\{t_ i\}^{\infty}_{i=0}$$ denotes any sequence of consecutive zeros of any solution z(t) of (1), $$\lambda >-1$$, $$\alpha <1+\lambda /2$$. Using the symbol $$f={\mathcal O}(t^{- \alpha})$$, $$t\to \infty$$ for the case when $$f=O(t^{-\alpha})$$ together with $$\lim_{t\to \infty}| f(t)| t^{\alpha +\xi}=\infty \text{ for } every\quad \xi >0,$$ the author proves the following theorem:
Let $$q(\infty)>0$$ and for $$k=0,1,...,n+2$$ let hold $$(-1)^ kq^{(k)}(t)\geq 0,\quad 0<t<\infty,\quad q^{(k)}={\mathcal O}(t^{- (k+\epsilon)}),\quad t\to \infty,\quad \epsilon >0.$$ Then (1) has a pair of solutions $$y_ 1(t)$$, $$y_ 2(t)$$ such that the function $$p(t)=y^ 2_ 1(t)+y^ 2_ 2(t)$$ satisfies p(t)$$\to 1$$ for $$t\to \infty$$, $$(- 1)^ kp^{(k+1)}(t)\geq 0,\quad \mu_ k<t<\infty,\quad k=0,1,...,n,$$ where $$\{\mu_ k\}^ n_ 1$$ is a nondecreasing sequence and $$\mu_ k=\mu_{k+1}$$ only if $$\mu_ k=0$$; $$p^{(k)}={\mathcal O}(t^{- (k+\epsilon)}),\quad t\to \infty,\quad n\geq 4,\quad k=1,2,....,n-3,$$ and the corresponding quantities $$M_ i$$ satisfy $$(-1)^ k\Delta^{k+1}M_ i\geq 0,\quad k=0,...,n-3,\quad i=l_ k,\quad l_ k+1,...,$$ where $$l_ k=l(k)$$ is integer, $$0=l_ 0\leq l_ 1\leq...\leq l_{n-3}$$ and $$l_ k=l_{k+1}$$ only if $$l_ k=0$$.
Reviewer: J.Kalas (Brno)
##### MSC:
 34C11 Growth and boundedness of solutions to ordinary differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 34A40 Differential inequalities involving functions of a single real variable 34A30 Linear ordinary differential equations and systems, general
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