Inequalities Involving Functions and Their Integrals and by D. S. Mitrinović, J. E. Pečarić, A. M. Fink (auth.)

By D. S. Mitrinović, J. E. Pečarić, A. M. Fink (auth.)

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Let f(x) be a continuous differentiable function on I = [0,1] with piecewise continuously differentiable derivative f'( x), and let II . III be a sup norm. A. Sharma and J. Tzimbalario [117] establish the following sharp inequalities. If IIfllI :::; 1, 1If" - ,\2 fill:::; A, ,\ > 0, then 11f' ± ,\flll :::; { ,\ + ,\ coth ~ + ,\ + 2VA, 1 tanh~, for 0 :::; A :::; ,\2 coth2~, for ,\2 coth ~ < A. Sharp estimates are also obtained for 1If'III if Ilflll :::; 1 and 11f" +,\2 fill:::; A. 61. Some integral inequalities are obtained by C.

Bollobas and J. R. Partington [198]. 98. T. Kato and 1. Satake [199] developed an algebraic theory of LandauKolmogorov inequalities. 99. Zvyaginstev [200] continues his investigation from [176]' [177] of LandauKolmogorov inequalities. 100. Landau-Kolmogorov type inequalities for functions on Rn to a Banach space are studied in Redheffer and Walter [201]. The domains of the functions must be restricted in order to get these types of inequalities. 101. If f is only defined on [a, b] then one can get inequalities of the Landau type 1If'1I2 ~ Kllfllllf"11 under further boundary conditions.

11) a] IIxll, b(a) d(a) = ~a II A3 x ll, = M(13+ a) IIA 3 xll. For fixed a > 1 the minimal values of the right hand side of the first (resp. 10) is attained at t = 2- 1/ 3a(a)1/3b(a)-1/3 (resp. t = 21/ 3c(a)1/3d(a)-1/3). 13) hold true for every a > 1. 13) we find that the minimal values of both functions are attained at some root of the cubic polynomial 3 6M 4M FM(a) = (a -1) - - - ( a -1) - - - . M+1 M+1 Now FM(l) < 0 and FM(l) < o. Furthermore, FM(a) has exactly one zero larger than 1. From this we see that FM has exactly one root which is greater than 1.

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