## On inverses of the Hölder inequality.(English)Zbl 0752.26009

The author derives an inequality complementary to $x^{1/p} y^{1/q}\leq x/p+y/q.$ Using this inequality, then converse Hölder- type inequalities of the form $a\| f\|_ 1+b\| g\|_ 1\leq c\| f^{1/p} g^{1/q}\|_ 1\qquad\text{and}\qquad \| f\|_ p\| g\|_ q\leq d\| fg\|_ 1$ are obtained, where the ranges of $$f$$ and $$g$$ are compact subintervals of $$(0,\infty)$$. The constants $$c$$ and $$d$$ then depend on $$a$$, $$b$$, $$p$$, $$q$$ and the endpoints of the intervals containing the ranges of $$f$$ and $$g$$.
The results include a lot of earlier inequalities complementary to Hölder’s one.

### MSC:

 26D15 Inequalities for sums, series and integrals
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### References:

 [1] Diaz, J.B.; Goldman, A.J.; Metcalf, F.T., Equivalence of certain inequalities complementary to those of Cauchy-Schwarz and Hölder, J. res. nat. bur. standards sect. B, 68, 147-149, (1964) · Zbl 0125.03001 [2] Diaz, J.B.; Metcalf, F.T., Complementary inequalities. II. inequalities complementary to the buniakowsky-Schwarz inequality for integrals, J. math. anal. appl., 9, 278-293, (1964) · Zbl 0135.34702 [3] Kantorovich, L.V., Functional analysis and applied mathematics, Uspekhi mat. nauk., 3, 89-185, (1948) · Zbl 0034.21203 [4] Mitrinović, D.S., Analytic inequalities, (1970), Springer-verlag Berlin · Zbl 0199.38101 [5] Nehari, Z., Inverse Hölder inequalities, J. math. anal. appl., 405-420, (1968) · Zbl 0182.38401 [6] Pólya, G.; Szegö, G.; Pólya, G.; Szegö, G.; Pólya, G.; Szegö, G., (), 214 [7] Rudin, W., Real and complex analysis, (1974), McGraw-Hill New York [8] Wang, C.L., Variants of the Hölder inequality and its inverses, Canad. math. bull., 20, 377-384, (1977) · Zbl 0398.26018 [9] Wang, C.-L., On development of inverses of the Cauchy and Hölder inequalities, SIAM rev., 21, 550-557, (1979) · Zbl 0414.26007
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