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On Landau type inequalities for functions with Hölder continuous derivatives. (English) Zbl 1060.26018

Suppose that \(I\) is a nondegenerate interval and \(f:I\to\mathbb{R}\) is a locally absolutely continuous function. Moreover, \(f\in L_{\infty}(I)\) and \(f^{\prime}\) satisfies Hölder’s condition \[ | f^{\prime}(t)-f^{\prime}(s)| \leq H| t-s| ^{\alpha} \] for all \(t,s\in I;\) here \(H>0\) and \(\alpha\in(0,1].\) It is proved that \(f^{\prime}\in L_{\infty}(I)\) and \[ \| f^{\prime}\| \leq H^{1/(\alpha+1)}[ 2( 1+1/\alpha) ] ^{\alpha/(\alpha+1)}\| f\| ^{\alpha/(\alpha+1)} \] if the length of \(I\) verifies \(\ell(I)\geq[ 2( 1+1/\alpha) ] ^{1/(\alpha+1)}( \| f\| /H) ^{1/(\alpha+1)}\), while \[ \| f^{\prime}\| \leq2\| f\| /\ell(I)+H\ell(I)^{\alpha}/(\alpha+1) \] if \(\ell(I)\leq[ 2( 1+1/\alpha) ] ^{1/(\alpha +1)}( \| f\| /H) ^{1/(\alpha+1)}\).

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26A16 Lipschitz (Hölder) classes
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