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Landau-Kolmogorov Constants

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Let ||f|| be the supremum of |f(x)|, a real-valued function f defined on (0,infty). If f is twice differentiable and both f and f^('') are bounded, Landau (1913) showed that

||f^'||<=2||f||^(1/2)||f^('')||^(1/2),
(1)

where the constant 2 is the best possible. Schoenberg (1973) extended the result to the nth derivative of f defined on (0,infty) if both f and f^((n)) are bounded,

||f^((k))||<=C(n,k)||f||^(1-k/n)||f^((n))||^(k/n).
(2)

An explicit formula for C(n,k) is not known, but particular cases are

C(3,1)=((243)/8)^(1/3)
(3)
C(3,2)=24^(1/3)
(4)
C(4,1)=4.288...
(5)
C(4,2)=5.750...
(6)
C(4,3)=3.708....
(7)

Let ||f|| be the supremum of |f(x)|, a real-valued function f defined on (-infty,infty). If f is twice differentiable and both f and f^('') are bounded, Hadamard (1914) showed that

||f^'||<=sqrt(2)||f||^(1/2)||f^('')||^(1/2),
(8)

where the constant sqrt(2) is the best possible. Kolmogorov (1962) determined the best constants C(n,k) for

||f^((k))||<=C(n,k)||f||^(1-k/n)||f^((n))||^(k/n)
(9)

in terms of the Favard constants

a_n=4/pisum_(j=0)^infty[((-1)^j)/(2j+1)]^(n+1)
(10)

by

C(n,k)=a_(n-k)a_n^(-1+k/n).
(11)

Special cases derived by Shilov (1937) are

C(3,1)=(9/8)^(1/3)
(12)
C(3,2)=3^(1/3)
(13)
C(4,1)=((512)/(375))^(1/4)
(14)
C(4,2)=sqrt(6/5)
(15)
C(4,3)=((24)/5)^(1/4)
(16)
C(5,1)=((1953125)/(1572864))^(1/5)
(17)
C(5,2)=((125)/(72))^(1/5).
(18)

For a real-valued function f defined on (-infty,infty), define

||f||=sqrt(int_(-infty)^infty[f(x)]^2dx).
(19)

If f is n differentiable and both f and f^((n)) are bounded, Hardy et al. (1934) showed that

||f^((k))||<=||f||^(1-k/n)||f^((n))||^(k/n),
(20)

where the constant 1 is the best possible for all n and 0<k<n.

For a real-valued function f defined on (0,infty), define

||f||=sqrt(int_0^infty[f(x)]^2dx).
(21)

If f is twice differentiable and both f and f^('') are bounded, Hardy et al. (1934) showed that

||f^'||<=sqrt(2)||f||^(1/2)||f^((n))||^(1/2),
(22)

where the constant sqrt(2) is the best possible. This inequality was extended by Ljubic (1964) and Kupcov (1975) to

||f^((k))||<=C(n,k)||f||^(1-k/n)||f^((n))||^(k/n)
(23)

where C(n,k) are given in terms of zeros of polynomials. Special cases are

C(3,1)=C(3,2)=3^(1/2)[2(2^(1/2)-1)]^(-1/3)
(24)
=1.84420...
(25)
C(4,1)=C(4,3)=sqrt((3^(1/4)+3^(-3/4))/a)
(26)
=2.27432...
(27)
C(4,2)=sqrt(2/b)=2.97963...
(28)
C(4,3)=((24)/5)^(1/4)
(29)
C(5,1)=C(5,4)=2.70247...
(30)
C(5,2)=C(5,3)=4.37800...,
(31)

where a is the least positive root of

x^8-6x^4-8x^2+1=0
(32)

and b is the least positive root of

x^4-2x^2-4x+1=0
(33)

(Franco et al. 1985, Neta 1980). The constants C(n,1) are given by

C(n,1)=sqrt(((n-1)^(1/n)+(n+1)^(-1+1/n))/c),
(34)

where c is the least positive root of

int_0^cint_0^infty(dxdy)/((x^(2n)-yx^2+1)sqrt(y))=(pi^2)/(2n).
(35)

An explicit formula of this type is not known for k>1.

The cases p=1, 2, infty are the only ones for which the best constants have exact expressions (Kwong and Zettl 1992, Franco et al. 1983).

See also

Landau Constant, Landau-Ramanujan Constant

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References

Finch, S. R. "Landau-Kolmogorov Constants." §3.3 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 212-216, 2003.Franco, Z. M.; Kaper, H. G.; Kwong, M. N.; and Zettl, A. "Bounds for the Best Constants in Landau's Inequality on the Line." Proc. Roy. Soc. Edinburgh 95A, 257-262, 1983.Franco, Z. M.; Kaper, H. G.; Kwong, M. N.; and Zettl, A. "Best Constants in Norm Inequalities for Derivatives on a Half Line." Proc. Roy. Soc. Edinburgh 100A, 67-84, 1985.Hadamard, J. "Sur le module maximum d'une fonction et de ses dérivés." Comptes Rendus Acad. Sci. Paris 41, 68-72, 1914.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities. Cambridge, England: Cambridge University Press, 1934.Kolmogorov, A. "On Inequalities Between the Upper Bounds of the Successive Derivatives of an Arbitrary Function on an Infinite Integral." Amer. Math. Soc. Translations, Ser. 1 2, 233-243, 1962.Kupcov, N. P. "Kolmogorov Estimates for Derivatives in L_2(0,infty)." Proc. Steklov Inst. Math. 138, 101-125, 1975.Kwong, M. K. and Zettl, A. Norm Inequalities for Derivatives and Differences. New York: Springer-Verlag, 1992.Landau, E. "Einige Ungleichungen für zweimal differentzierbare Funktionen." Proc. London Math. Soc. Ser. 2 13, 43-49, 1913.Landau, E. "Die Ungleichungen für zweimal differentzierbare Funktionen." Danske Vid. Selsk. Math. Fys. Medd. 6, 1-49, 1925.Ljubic, J. I. "On Inequalities Between the Powers of a Linear Operator." Amer. Math. Soc. Trans. Ser. 2 40, 39-84, 1964.Neta, B. "On Determinations of Best Possible Constants in Integral Inequalities Involving Derivatives." Math. Comput. 35, 1191-1193, 1980.Schoenberg, I. J. "The Elementary Case of Landau's Problem of Inequalities Between Derivatives." Amer. Math. Monthly 80, 121-158, 1973.Shilov, G. E. "On Inequalities Between Derivatives." Sbornik Rabot Studencheskikh Nauchnykh Kruzhov Moskovskogo Gosudarstvennogo Universiteta, 17-27, 1937.

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Landau-Kolmogorov Constants

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Weisstein, Eric W. "Landau-Kolmogorov Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Landau-KolmogorovConstants.html

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