Using the variational characterization of principal (i.e., smallest) eigenvalue below essential spectrum a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing finite interval length) Friedrichs extension $$ T_F$$ minimal operator for regular four-coefficient Sturm–Liouville differential expressions. In more general singular context, these expre...

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The interaction between light and quantum-dot (QD) excitons is strongly influenced by the environment in which the QD is placed. We have investigated the interaction by measuring the time-resolved spontaneous-emission rate of QD excitons in different nanostructured environments. Thereby, we have determined the oscillator strength, quantum efficiency and spin-flip rates of QD excitons as well as...

and Applied Analysis 3 Theorem 1.6 5 , Theorem 4 . Letf : I → R0 be differentiable on I◦, a, b ∈ I, a < b, and f ′ ∈ L a, b . If |f ′ x | is s-convex on a, b for some s ∈ 0, 1 , and p, q ≥ 1, such that 1/q 1/p 1 then ∣ ∣ ∣ ∣ ∣ f ( a b 2 ) − 1 b − a ∫b a f x dx ∣ ∣ ∣ ∣ ∣ ≤ 2−1/p ∣f ′ a ∣ ∣q s 1 ∣ ∣f ′ a b /2 ∣ ∣q)1/q { s 1 s 2 } 2−1/p ∣f ′ b ∣ ∣q s 1 ∣ ∣f ′ a b /2 ∣ ∣q)1/q { s 1 s 2 } 2−1/p [( β...

and Applied Analysis 3 where ‖ · ‖p p ∈ 1,∞ are the Lebesgue norms, that is, ∥∥f ′∥∞ ess sup s∈ a,b ∣∣f ′ s ∣∣, ∥∥f ′∥∥ p : (∫b a ∣∣f ′ s ∣∣ds)1/p, p ≥ 1. 1.6 The case of convex functions is as follows 4 . Theorem 1.5. Let f : a, b → be a convex function on a, b . Then one has the inequalities 1 8 b − a 2 [ f ′ ( a b 2 ) − f ′ − ( a b 2 )] ≤ f a f b 2 b − a − ∫b