Abstract
The Hermite-Hadamard inequality is an inequality for convex functions that gives an estimate for the integral mean value of a convex function on a closed interval by its value at the middle of interval and the average of its values at the endpoints. The Hermite-Hadamard inequality can be generalized by using the Riemann-Stieltjes integral mean value. An application of the Hermite-Hadamard inequality with respect to Riemann-Stieltjes integral for estimating the power mean of positive real numbers by the aritmethic mean is given at the end of discussion.
Highlights
Pembuktian ketaksamaan (1) dan (2) dapat dilihat di (Roberts dan Varberg, 1975)
The Hermite-Hadamard inequality is an inequality for convex functions that gives an estimate for the integral mean value of a convex function on a closed interval by its value at the middle of interval and the average of its values at the endpoints
The Hermite-Hadamard inequality can be generalized by using the RiemannStieltjes integral mean value
Summary
Sebagai akibat dari teorema ini adalah fungsi f memiliki turunan sepihak pada (a, b) dan untuk setiap x, y ∈ (a, b) yang memenuhi x < y berlaku f−′(x) f+′ (x) f(y) y f(x) x f−′(y) ≤ f+′(y). Ternyata dengan meninjau nilai rata-rata integral Riemann-Stieltjes terhadap fungsi g yang monoton naik pada [a, b] dan g (b) − g (a) = 1, ketaksamaan (4) dapat diperumum menjadi ketaksamaan Hermite-Hadamard terhadap integral RiemannStieltjes. Karena fungsi g monoton naik pada [a, b], maka untuk setiap g(x) − g(a) ≥ 0 dan g(b) − g(x) ≥ 0. Karena f adalah fungsi konveks, maka berdasarkan lema 2, untuk setiap x ∈ [a, b] berlaku f(b) − f(a) f(x) ≤ f(a) + b − a (x − a) Dengan mengintegralkan kedua ruas ketaksamaan di atas dan membaginya dengan g(b) − g(a) diperoleh f(a) f(b) b f(a) a (xg a). Aplikasi ketaksamaan (5) dengan pemilihan fungsi g tertentu diberikan pada bagian selanjutnya
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