2025/07/14 by Goswami, Angshuman R.
Computer Science · Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Equations Stability Results #General Mathematics (math.GM) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2508.00854
openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main objective of this paper is to present Ostrowski's inequality for a broader class of functions and to propose a refinement to the classical version of it. The original Ostrowski's inequality can be stated as follows "If f:[a,b]→ℝ is differentiable and f'∈ L∞[a, b], then for any p∈ ]a,b[ , the following functional inequality holds: |f(p)-\dfrac1b-a∫abf(t) dt|≤ \dfrac(p-a)2+(b-p)22(b-a)‖ f'‖ab . ^^^^" We relax the condition of differentiability and show that even if f∈ C[a,b] is non-differentiable at the points p_1,⋯,p_n, then for any p∈ ]a,b[ ∖\oversetn\underseti=1∪\p_i\, the following Ostrowski-type inequality holds: | f(p) - (1)/(b-a) ∫ab f(t) dt | ≤ (1)/(2) max \ amp; ‖ f' ‖ap1(p1 - a), …, ‖ f' ‖_pi-1p (p - pi-1), ‖ f' ‖ppi (pi - p),
amp; …, ‖ f' ‖pnb (b - pn) \ + max \ f(a) + ∑i=1n f(pi), -∑i=1n f(pi) - f(b) \. Also, we investigate the possibility of proposing a refinement for Ostrowski inequality. We prove that if f'∈ L∞[a, b], then for any p∈ ]a,b[, we can restructure the inequality as follows: | f(p) - (1)/(b-a) ∫ab f(t) dt | ≤ min \ amp; [ (1)/(4) + ( (p - (a + b)/(2))/(b - a) )2 ](b - a) ‖ f' ‖ab,
amp; + (1)/(2) max \ (p - a) ‖ f' ‖ap, (b - p) ‖ f' ‖pb \ \.