[Paper Review] New Inequalities for Hermite-Hadamard and Simpson Type and Applications

M. Emіn Özdemіr, Çetin Yıldız|arXiv (Cornell University)|Mar 10, 2011

Mathematical Inequalities and Applications3 citations

TL;DR

This paper establishes new inequalities of Hermite-Hadamard and Simpson types for functions whose second derivatives are P-convex, using integral identities and power mean inequalities. The key contribution is tighter error bounds for quadrature rules and applications to special means of real numbers, improving upon existing results under P-convexity assumptions on |f''|.

ABSTRACT

In this paper, we obtain new bounds for the inequalities of Simpson and Hermite-Hadamard type for functions whose second derivatives absolute values are P-convex. These bounds can be much better than some obtained bounds. Some applications for special means of real numbers are also given.

Motivation & Objective

• To derive tighter error bounds for Hermite-Hadamard and Simpson-type inequalities under P-convexity of |f''|.
• To generalize existing integral inequalities using a parameterized integral identity involving f'' and kernel functions.
• To apply the new bounds to special means of real numbers, such as arithmetic, logarithmic, and generalized log-mean.
• To compare the new bounds with classical results and demonstrate cases where they are sharper.

Proposed method

• Utilizes a parameterized integral identity (Lemma 1) linking the difference between integral averages and function values to the second derivative via a kernel function k(t).
• Applies P-convexity of |f''| to majorize integrals over [0, 1/2] and [1/2, 1] using |f''(a)| and |f''(b)|.
• Employs Hölder’s inequality with conjugate exponents p and q to derive L^q-type bounds on the error terms.
• Derives explicit bounds depending on λ ∈ [0,1], distinguishing cases for λ ≤ 1/2 and λ ≥ 1/2.
• Applies the results to f(x) = x^n to obtain inequalities involving arithmetic, logarithmic, and generalized log-mean functions.
• Compares the new bounds with classical inequalities (e.g., (1.3), (1.4)) and identifies conditions under which they are tighter.

Experimental results

Research questions

• RQ1 Can tighter bounds be established for Hermite-Hadamard and Simpson-type inequalities when |f''| is P-convex rather than bounded?
• RQ2 How does the parameter λ in the integral identity affect the resulting error estimates for different types of quadrature rules?
• RQ3 Under what conditions do the new bounds improve upon classical inequalities like (1.3) and (1.4)?
• RQ4 Can the new inequalities be effectively applied to derive bounds for special means of real numbers?
• RQ5 What is the role of the L^q norm of |f''| in deriving sharp error estimates under P-convexity?

Key findings

• For λ ∈ [0, 1/2], the error bound is ≤ (b−a)²/24 × (8λ³ − 3λ + 1)(|f''(a)| + |f''(b)|), which improves upon classical bounds when |f''| is P-convex.
• For λ ∈ [1/2, 1], the bound is ≤ (b−a)²/24 × (3λ − 1)(|f''(a)| + |f''(b)|), providing a tighter estimate than (1.3) when K < M.
• When λ = 1, the bound becomes ≤ (b−a)²/24 × (|f''(a)|^q + |f''(b)|^q)^{1/q}, which reduces to (1.4) under boundedness of |f''|.
• For λ = 1/3, the Simpson-type bound is ≤ (b−a)²/162 × (|f''(a)|^q + |f''(b)|^q)^{1/q}, improving upon classical (1.2) under P-convexity.
• When |f''| ≤ M, the bounds reduce to ≤ M(b−a)²/24 × 2^{1/q}, which is tighter than (1.3) when K < M.
• Applications to f(x) = x^n yield explicit inequalities between A(a,b), L_n(a,b), and A^n(a,b), valid for |n(n−1)| ≥ 3 and q ≥ 1.