Subtopic Deep Dive
Hermite-Hadamard Inequalities
Research Guide
What is Hermite-Hadamard Inequalities?
Hermite-Hadamard inequalities state that for a convex function f on [a,b], f((a+b)/2) ≤ (1/(b-a)) ∫_a^b f(x) dx ≤ (f(a)+f(b))/2.
These inequalities provide bounds relating function values at endpoints and midpoint to average integral values. Extensions apply to s-convex, harmonically convex, quantum, and fractional integral settings. Dragomir and Pearce (2003) survey applications with 598 citations; Işcan (2014) introduces harmonically convex cases with 286 citations.
Why It Matters
Hermite-Hadamard inequalities yield error estimates for midpoint quadrature rules in numerical integration (Dragomir and Pearce, 2003). They bound approximations in fractional calculus for solving differential equations (Sarıkaya and Yıldırım, 2017; Dahmani, 2010). Quantum versions support estimates in quantum calculus applications (Tariboon and Ntouyas, 2014; Noor et al., 2014).
Key Research Challenges
Fractional integral extensions
Proving Hermite-Hadamard inequalities for Riemann-Liouville fractional integrals requires new identities to handle non-integer orders. Sarıkaya and Yıldırım (2017) establish such inequalities using fractional identities (239 citations). Challenges persist for higher-order fractional operators.
Quantum and q-analogues
Adapting inequalities to quantum calculus and q-deformations demands revised convexity notions. Alp et al. (2016) derive q-Hermite-Hadamard inequalities for convex functions (229 citations). Verifying monotonicity in quantum settings remains open (Tariboon and Ntouyas, 2014).
Harmonic convexity refinements
Extending to harmonically convex functions involves reciprocal transformations complicating proofs. Işcan (2014) proves type inequalities for these functions (286 citations). Refinements for generalized harmonic classes face sharpness issues.
Essential Papers
Selected Topics on Hermite-Hadamard Inequalities and Applications
Sever S Dragomir, Charles Pearce · 2003 · SSRN Electronic Journal · 598 citations
The Hermite-Hadamard double inequality is the first fundamental result for convex functions defined on a interval of real numbers with a natural geometrical interpretation and a loose number of app...
HERMITE-HADAMARD TYPE INEQUALITIES FOR HARMONICALLY CONVEX FUNCTIONS
İmdat Işcan · 2014 · Hacettepe Journal of Mathematics and Statistics · 286 citations
The author introduces the concept of harmonically convex functions and establishes some Hermite-Hadamard type inequalities of these classes of functions
On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals
Mehmet Zeki Sarıkaya, Hüseyin Yıldırım · 2017 · Miskolc mathematical notes/Mathematical notes · 239 citations
In this paper, we have established Hermite-Hadamard-type inequalities for fractional integrals and will be given an identity.With the help of this fractional-type integral identity, we give some in...
q -Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions
Necmettin Alp, Mehmet Zeki Sarıkaya, Mehmet Kunt et al. · 2016 · Journal of King Saud University - Science · 229 citations
In this paper, we prove the correct q-Hermite–Hadamard inequality, some new q-Hermite–Hadamard inequalities, and generalized q-Hermite–Hadamard inequality. By using the left hand part of the correc...
On Minkowski and Hermite-Hadamard integral inequalities via fractional integration
Zoubir Dahmani · 2010 · Annals of Functional Analysis · 226 citations
Quantum integral inequalities on finite intervals
Jessada Tariboon, Sotiris K. Ntouyas · 2014 · Journal of Inequalities and Applications · 200 citations
In this paper, some of the most important integral inequalities of analysis are extended to quantum calculus. These include the Hölder, Hermite-Hadamard, trapezoid, Ostrowski, Cauchy-Bunyakovsky-Sc...
On q-Hermite–Hadamard inequalities for general convex functions
Sergio Bermudo, Péter Kórus, Juan E. Nápoles Valdés · 2020 · Acta Mathematica Academiae Scientiarum Hungaricae · 196 citations
Reading Guide
Foundational Papers
Start with Dragomir and Pearce (2003, 598 citations) for core inequality and applications; Işcan (2014, 286 citations) for harmonic convexity; Dahmani (2010, 226 citations) for fractional integrals.
Recent Advances
Sarıkaya and Yıldırım (2017, 239 citations) on fractional types; Bermudo et al. (2020, 196 citations) on general q-convex; Alp et al. (2016, 229 citations) for quantum midpoint estimates.
Core Methods
Convexity via Jensen functional bounds (Dragomir, 2006); fractional identities (Sarıkaya 2017); q-deformations and quantum integrals (Alp 2016, Tariboon 2014).
How PapersFlow Helps You Research Hermite-Hadamard Inequalities
Discover & Search
Research Agent uses searchPapers('Hermite-Hadamard fractional integrals') to find Sarıkaya and Yıldırım (2017, 239 citations), then citationGraph to map 50+ extensions from Dragomir and Pearce (2003). exaSearch uncovers niche q-analogues; findSimilarPapers links Işcan (2014) to harmonic variants.
Analyze & Verify
Analysis Agent applies readPaperContent on Alp et al. (2016) to extract q-Hermite proofs, then runPythonAnalysis to plot convexity bounds with NumPy: verifyResponse(CoVe) checks inequality sharpness via GRADE scoring. Statistical verification confirms monotonicity in quantum estimates from Noor et al. (2014).
Synthesize & Write
Synthesis Agent detects gaps in fractional-quantum overlaps, flags contradictions between Işcan (2014) and Sarıkaya (2017); Writing Agent uses latexEditText for proofs, latexSyncCitations to bibtex Dragomir (2003), latexCompile for arXiv-ready paper, exportMermaid for inequality flowcharts.
Use Cases
"Plot error bounds for Hermite-Hadamard in midpoint rule using Python."
Research Agent → searchPapers → Analysis Agent → runPythonAnalysis(NumPy plot of Dragomir 2003 bounds) → matplotlib figure verifying (f(mid) - integral avg) ≤ error.
"Write LaTeX proof of q-Hermite inequality for convex functions."
Research Agent → findSimilarPapers(Alp 2016) → Synthesis → gap detection → Writing Agent → latexEditText(proof) → latexSyncCitations → latexCompile → PDF with theorem environment.
"Find GitHub repos implementing quantum Hermite-Hadamard estimates."
Code Discovery → paperExtractUrls(Tariboon 2014) → paperFindGithubRepo → githubRepoInspect → exportCsv of quantum integral code snippets for NumPy verification.
Automated Workflows
Deep Research workflow scans 50+ papers via citationGraph from Dragomir (2003), structures report on fractional extensions (Sarıkaya 2017). DeepScan applies 7-step CoVe to verify Işcan (2014) harmonic proofs with GRADE checkpoints. Theorizer generates conjectures for unified q-quantum Hermite bounds from Alp (2016) and Noor (2014).
Frequently Asked Questions
What is the classical Hermite-Hadamard inequality?
For convex f on [a,b], it states f((a+b)/2) ≤ (1/(b-a)) ∫_a^b f ≤ (f(a)+f(b))/2. Dragomir and Pearce (2003) provide geometrical interpretation and applications (598 citations).
What methods extend it to fractional integrals?
Riemann-Liouville fractional identities enable proofs; Sarıkaya and Yıldırım (2017) derive inequalities via integral identities (239 citations). Dahmani (2010) applies to Minkowski types (226 citations).
What are key papers on quantum Hermite-Hadamard?
Tariboon and Ntouyas (2014) extend to quantum integrals (200 citations); Noor et al. (2014) give estimates (186 citations). Alp et al. (2016) cover q-analogues (229 citations).
What open problems exist?
Unified bounds for mixed fractional-quantum convexity; refinements for s-harmonic classes beyond Işcan (2014). Sharpness in q-settings from Bermudo et al. (2020) remains unresolved.
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