id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-5694	Abbasi, Anjum Mustafa Khan; Matloob Anwar	Analysis of Hardy-type Inequalities Involving Green Functions and Taylor’s Polynomial Approximation	2025	18	.pdf	application/pdf	5869	366	76	Math, 18 (1) (2025), 5694 3 of 18 + ∫ a2 a1 G1(ϖ, τ)F ′′′(τ)dτ, (1) F(ϖ) = F(a2) + (ϖ − a2)F ′(a1) + (ϖ − a1)(ϖ − a2)F ′′(a2)− (ϖ − a2) 2 2 F ′′(a1) − ∫ a2 a1 G2(ϖ, τ)F ′′′(τ)dτ, (2) F(ϖ) = F(a2) + (ϖ − a1)F ′(a1) + (a2 − a1)F ′(a2)− [ (ϖ − a1) 2 2 + (ϖ − a1)(a− b) ] F ′′(a1) + [ (a2 − a1) 2 2 + (ϖ − a1)(ϖ − a2) ] F ′′(a2)− ∫ a2 a1 G3(ϖ, τ)F ′′′(τ)dτ (3) and F(ϖ) = F(a2) + (a2 − a1)F ′(a1) + (b−ϖ)F ′(a2) + [ (a2 − a1) 2 2 + (ϖ − a2)(ϖ − a1) ] F ′′(a1) − [ (ϖ − a2) 2 2 + (ϖ − a2)(a2 − a1) ] F ′′(a2) = 1 (n− 4)! (∫ a2 a1 B1(τ)F (n)(τ)dτ − F (n−1)(a2)−F (n−1)(a1) (a2 − a1) ∫ a2 a1 B1(τ)dτ ) .(44)	cache/ejpam-5694.pdf	txt/ejpam-5694.txt