Sentence examples for valid for all functions from inspiring English sources

The limit inequality is valid for all functions in the Sobolev space H1.

Such assertions are contained in Lemmas 9 and 10 and it is important that they are valid for all functions in B ¯ and not only for solutions of problem (1), (2).

These considerations indicate that the frame region in Theorem 1.2 in a quite accurate way describes the maximally possible frame set below (b=2) that is valid for all the functions in (V_{N,a}), at least for (N=2).

This formula is valid for all arbitrary function u ∈ K R + ⊆ D + which is a continuously differentiable function of order ( 2 m + 2 ).

They are valid for any function that can be expressed by a power series.

Our work is quite general in nature as the bounds expressed in terms of m(t) and σ(t) units and the bounds are valid for all parent distribution functions F such that F-1 t)=p,0<p<1. F-1 t

Elezović, Giordano and Pečarić [17] established the double inequality biggl(frac{1}{2}+sqrt{frac{1}{4}+x} biggr)^{1-x}x^{x}< Gamma (x+1)< 2^{1-x}x^{x} (1.2) for the gamma function being valid for all (xin 0, 1)), and asked for 'other bounds for the gamma function in terms of elementary functions'.

However, Theorem A does not remain valid for meromorphic functions.

Theorem 2.1 is still valid for convex functions.

Heittokangas et al. [9], prove the following result which is a shifted analogue of Brück conjecture valid for meromorphic functions.

The theory is valid for general functions as long as they satisfy the qualitative properties above.