Landau-Kolmogorov Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $\vert\vert f\vert\vert$ be the Supremum of $\vert f(x)\vert$, 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

$$\vert\vert f'\vert\vert\leq 2\vert\vert f\vert\vert^{1/2} \vert\vert f''\vert\vert^{1/2},$$ (1)

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

$$\vert\vert f^{(k)}\vert\vert\leq C(n,k)\vert\vert f\vert\vert^{1-k/n}\vert\vert f^{(n)}\vert\vert^{k/n}.$$ (2)

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

$$C(3,1) = \left({243\over 8}\right)^{1/3}$$ (3)

$$C(3,2) = 24^{1/3}$$ (4)

$$C(4,1) = 4.288\ldots$$ (5)

$$C(4,2) = 5.750\ldots$$ (6)

$$C(4,3) = 3.708\ldots.$$ (7)

Let $\vert\vert f\vert\vert$ be the Supremum of $\vert f(x)\vert$, 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

$$\vert\vert f'\vert\vert\leq \sqrt{2} \vert\vert f\vert\vert^{1/2} \vert\vert f''\vert\vert^{1/2},$$ (8)

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

$$\vert\vert f^{(k)}\vert\vert\leq C(n,k)\vert\vert f\vert\vert^{1-k/n}\vert\vert f^{(n)}\vert\vert^{k/n}$$ (9)

in terms of the Favard Constants

$$a_n={4\over\pi} \sum_{j=0}^\infty \left[{(-1)^j\over 2j+1}\right]^{n+1}$$ (10)

by

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

Special cases derived by Shilov (1937) are

$$C(3,1) = \left({9\over 8}\right)^{1/3}$$ (12)

$$C(3,2) = 3^{1/3}$$ (13)

$$C(4,1) = \left({512\over 375}\right)^{1/4}$$ (14)

$$C(4,2) = \sqrt{6\over 5}$$ (15)

$$C(4,3) = \left({24\over 5}\right)^{1/4}$$ (16)

$$C(5,1) = \left({1953125\over 1572864}\right)^{1/5}$$ (17)

$$C(5,2) = \left({125\over 72}\right)^{1/5}.$$ (18)

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

$$\vert\vert f\vert\vert=\sqrt{\int_{-\infty}^\infty [f(x)]^2\,dx}\,.$$ (19)

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

$$\vert\vert f^{(k)}\vert\vert\leq \vert\vert f\vert\vert^{1-k/n} \,\vert\vert f^{(n)}\vert\vert^{k/n},$$ (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

$$\vert\vert f\vert\vert=\sqrt{\int_0^\infty [f(x)]^2\,dx}.$$ (21)

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

$$\vert\vert f'\vert\vert\leq \sqrt{2} \,\vert\vert f\vert\vert^{1/2} \vert\vert f^{(n)}\vert\vert^{1/2},$$ (22)

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

$$\vert\vert f^{(k)}\vert\vert\leq C(n,k)\,\vert\vert f\vert\vert^{1-k/n}\vert\vert f^{(n)}\vert\vert^{k/n}$$ (23)

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

$$C(3,1) = C(3,2)=3^{1/2} [2(2^{1/2}-1)]^{-1/3}$$

$$= 1.84420\ldots$$ (24)

$$C(4,1) = C(4,3)=\sqrt{3^{1/4}+3^{-3/4}\over a}$$

$$= 2.27432\ldots$$ (25)

$$C(4,2) = \sqrt{2\over b}=2.97963\ldots$$ (26)

$$C(4,3) = \left({24\over 5}\right)^{1/4}$$ (27)

$$C(5,1) = C(5,4)=2.70247\ldots$$ (28)

$$C(5,2) = C(5,3)=4.37800\ldots,$$ (29)

where $a$ is the least Positive Root of

$$x^8-6x^4-8x^2+1=0$$ (30)

and $b$ is the least Positive Root of

$$x^4-2x^2-4x+1=0$$ (31)

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

$$C(n,1)=\sqrt{(n-1)^{1/n}+(n+1)^{-1+1/n}\over c},$$ (32)

where $c$ is the least Positive Root of

$$\int_0^c \int_0^\infty {dx\,dy\over(x^{2n}-yx^2+1)\sqrt{y}}={\pi^2\over 2n}.$$ (33)

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).

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lk/lk.html

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.

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.