Abstract:
This thesis is a detailed study of convex functions, focusing on h-convex, p
convex, and stronglyireciprocally (p,h)-convex functions of higheriorder cases.iThe
mainiobjective ofithis research isito enhance both theiunderstanding and analysis of
these generalizediconvex functions, with a focus on their connections to established
mathematicaliinequalities, such as theiHermite-Hadamardiinequality, Fejériinequality,
and fractionaliintegraliinequalities. The foundational concept of convexity, its basic
properties, and applications are reexamined. The utility of convex functions in
numerous areas ranging from theoretical research to practical applications of pure
and appliedisciences including, economic models, mathematical analysis, and opti
mizationiproblems is highlighted. The role of inequality theory is also pointed out
for convex functions. The classical convexity is extended to stronglyireciprocally
(p, h)-convex functions of higheriorder instances, enabling a deeper insight into how
convex functions can be applied to mathematicaliinequalities, suchias theiHermite
Hadamardiinequality, the Fejériinequality, andifractional integraliinequalities are
explored. A novel mapping Mw
F(x) for h-convex functions is explored along a series
of useful results. These results are formulated through lemmas and propositions,
which are then utilized to derive some generalized Fejér-typeiinequalities and an
improved variant of Hermite-Hadamard inequalities. A second mapping, Hw
F(x) for h-convex function is introduced, with the further contribution of some results that
leads to the derivation of a significant theorem. These findings contribute new tools
for mathematical analysis and significantly broaden the scope of known results to
generalize convex functions and inequalities.
