A K-R defect framework for classical inequalities: curvature recovery, uniqueness, stability, and applications to nonlinear boundary value problems
- 1Independent Researcher
Abstract
We develop a K – R K−−R defect framework for classical inequalities centred on the normalised defect functional Φ f ( x , y ; K , R ) = D f / ( K R ( x − y ) 2 ) Φ f(x,y;K,R) = Df/(KR(x−y)2) , where D f ( x , y ; K , R ) = K f ( x ) + R f ( y ) − f ( K x + R y ) Df(x,y;K,R) = Kf(x)+Rf(y)−f(Kx+Ry) for an admissible pair satisfying K ≥ 0 K≥ 0 , R ≥ 0 R≥ 0 , K + R = 1 K+R=1 . The framework establishes a complete forward-inverse curvature theory. The Curvature Recovery Theorem proves that Φ f Φ f equals 1 2 f ″ ( ξ ) 1/2f''(ξ ) for some interior point ξ, making Φ f Φ f a derivative-free curvature estimator. The Local Limit Theorem establishes exact pointwise curvature recovery in the limit, with an explicit error bound. The Uniqueness Principle proves that two C 2 C2 functions with identical normalised defect fields differ by at most an affine function. The Constant Defect Characterisation identifies quadratic polynomials as precisely the C 2 C2 functions with constant normalised defect. The Reconstruction Formula recovers f from its defect measurements by double integration. The K – R K−−R Stability Theorem proves that a C 2 C2 function with small defect magnitude satisfying homogeneous boundary conditions is uniformly close to zero, with an explicit constant. A K – R K−−R deficit theory for the Hölder inequality is established, including a new K – R K−−R Isoperimetric Deficit Ratio, with equality conditions strictly stronger than the classical case. Applications to the nonlinear boundary value problem − u ″ = f ( u ) −u''=f(u) include derivative-free source term curvature bounds, a polynomial reconstruction algorithm with a proved positive-definite Hessian guarantee, and a derivative-free convexity diagnostic for solution profiles. A multivariable extension connects Φ f Φ f to the Rayleigh quotient of the Hessian, motivating a programme for full Hessian recovery from defect measurements.
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BibTeX
@article{pasupuleti2026defect,
title = {{A K-R defect framework for classical inequalities: curvature recovery, uniqueness, stability, and applications to nonlinear boundary value problems}},
author = {RamaKrishna Pasupuleti},
journal = {Boundary Value Problems},
year = {2026},
month = jul,
doi = {10.1186/s13661-026-02333-6},
url = {https://doi.org/10.1186/s13661-026-02333-6},
note = {CC BY},
}
RIS
TY - JOUR
TI - A K-R defect framework for classical inequalities: curvature recovery, uniqueness, stability, and applications to nonlinear boundary value problems
AU - RamaKrishna Pasupuleti
PY - 2026
DA - 2026/07/01
JO - Boundary Value Problems
DO - 10.1186/s13661-026-02333-6
UR - https://doi.org/10.1186/s13661-026-02333-6
AB - We develop a K – R K−−R defect framework for classical inequalities centred on the normalised defect functional Φ f ( x , y ; K , R ) = D f / ( K R ( x − y ) 2 ) Φ _f(x,y;K,R) = D_f/(KR(x−y)^2) , where D f ( x , y ; K , R ) = K f ( x ) + R f ( y ) − f ( K x + R y ) D_f(x,y;K,R) = Kf(x)+Rf(y)−f(Kx+Ry) for an admissible pair satisfying K ≥ 0 K≥ 0 , R ≥ 0 R≥ 0 , K + R = 1 K+R=1 . The framework establishes a complete forward-inverse curvature theory. The Curvature Recovery Theorem proves that Φ f Φ _f equals 1 2 f ″ ( ξ ) 1/2f''(ξ ) for some interior point ξ, making Φ f Φ _f a derivative-free curvature estimator. The Local Limit Theorem establishes exact pointwise curvature recovery in the limit, with an explicit error bound. The Uniqueness Principle proves that two C 2 C^2 functions with identical normalised defect fields differ by at most an affine function. The Constant Defect Characterisation identifies quadratic polynomials as precisely the C 2 C^2 functions with constant normalised defect. The Reconstruction Formula recovers f from its defect measurements by double integration. The K – R K−−R Stability Theorem proves that a C 2 C^2 function with small defect magnitude satisfying homogeneous boundary conditions is uniformly close to zero, with an explicit constant. A K – R K−−R deficit theory for the Hölder inequality is established, including a new K – R K−−R Isoperimetric Deficit Ratio, with equality conditions strictly stronger than the classical case. Applications to the nonlinear boundary value problem − u ″ = f ( u ) −u''=f(u) include derivative-free source term curvature bounds, a polynomial reconstruction algorithm with a proved positive-definite Hessian guarantee, and a derivative-free convexity diagnostic for solution profiles. A multivariable extension connects Φ f Φ _f to the Rayleigh quotient of the Hessian, motivating a programme for full Hessian recovery from defect measurements.
SN - 1687-2770
SN - 1687-2762
ER -