Let be an interval and a convex and differentiable function on (the interior of whose derivative is continuous on If is a selfadjoint operator on the Hilbert space with then (2.1).
Let be an interval and a convex and differentiable function on (the interior of whose derivative is continuous on. If is a selfadjoint operator on the Hilbert space with then (3.2). for any with.
Theorem 1.1 Let f : [ a, b ] → R be a differentiable mapping on ( a, b ) whose derivative is bounded on ( a, b ) and denote ∥ f ′ ∥ ∞ = sup t ∈ ( a, b ) | f ′ ( t ) | < ∞.
In 1938, Ostrowski proved the following important inequality [1]: Theorem 1.1 Let f : [ a, b ] → R be continuous on [ a, b ] and differentiable on ( a, b ) whose derivative f ′ : ( a, b ) → R is bounded on ( a, b ), i.e., ∥ f ′ ∥ ∞ = sup t ∈ ( a, b ) | f ′ ( t ) | < ∞.
It was shown in [42] that there exists a unique real function (xi ) in (L^1({mathbb R})) such that begin{aligned} {text {trace}}(f(B -f(A))=int _{mathB -f}f'(s)xi (s),ds, end{Aligned} (1.6.1)whenever f is a differentiable function on ({mathbb R}) whose derivative is the Fourier transform of an (L^1) function.
Let C Open image in new windowbe a curve parametrised in a neighbourhood of the singular point z(0) = 0 by z(t) = t + ı y(t) with |t| ≤ ε, where y(t) is a continuous function on the interval whose derivative is continuous away from zero in and bounded in this interval.
Let X be a reflexive real Banach space and (Isubseteq mathbb{R}) be an interval; let (Phi :Xto mathbb{R}) be a sequentially weakly lower semi-continuous (mathrm{C}^{1} -functional, bounded on each bounded subset of X, whose derivative admits a continuous inverse on (X^); let (J:Xto mathbb{R}) be a (mathrm{C}^{1} -functional with compact derivative.
Lemma 3.9 Let f : [ a, b ] → R be continuous on [ a, b ] and differentiable in ( a, b ), whose derivative is bounded on ( a, b ) and let ∥ f ′ ∥ ∞ = sup t ∈ ( a, b ) | f ′ ( t ) | < ∞.
Theorem 1.2 Suppose f : [ a, b ] → R is an absolutely continuous mapping on [ a, b ] whose derivative belongs to L p [ a, b ].
