

Title of Thesis
Exponential Convexity And Cauchy Means Introduced
By Inequalities For Monotone And Related Functions 
Author(s)
Atiq Ur Rehman 
Institute/University/Department
Details Abdus Salam School of Mathematical Sciences / GC
University, Lahore 
Session 2011 
Subject Mathematics 
Number of Pages 101 
Keywords (Extracted from title, table of contents and
abstract of thesis) Related, Demonstrate, Erences,
Means, Functions, Inequalities, Exponential, Functions, Comparison,
Literature, Monotone, Introduced, Convexity, Cauchy 
Abstract There is a lot of
literature available on convexity of functions. In contrast, the
literature on the exponential convexity is hardly available as there
is no operative criteria to recognize exponential convexity.It is
not easy to nd and construct exponentially convex functions
eventhough it is very important subclass of convex functions in
many ways. For example, Laplace transform of a nonnegative nite
measure is an exponentially convex function. Moreover, one can
derive results about positive de nite functions from the properties
of exponentially convex functions.
We consider the di erences of Petrovi c and related inequalities,
Giaccardi and related inequalities, Chebyshev's inequality,
inequality introduced by Lupa s and inequality introduced by LevinSteckin
to construct positive semide nite matrices.We derive the classes of
exponentially convex functions for the di erences and discuss their
properties. We introduce Cauchy means and prove the monotonicity of
these means by using the important property of exponentially convex
functions.As an application, we establish the mean value theorem of
Cauchy type.
In the rst chapter, we organize some basic notions and results.In
the second chapter, we use the JensenPetrovi c's inequality for
starshaped functions, generalized Petrovi c inequality and
inequality introduced by Vasi c and Pe cari c for increasing
functions to give results related to power sums.We consider the di
erence of these inequalities to construct positive semide nite
matrices for certain classes of functions to derive families of
exponential and logarithmic convex functions. We introduce new means
of Cauchy type related to power sums and establish comparison
between them.We, also illustrate integral analogs for some results
and prove related mean value theorems of Cauchy type.
In the third chapter, we prove the Giaccardi's type inequality for
starshaped type functions and the Giaccardi's inequality for
convexconcave antisymmetric functions. We assume the di erences of
Giaccardi's type inequality, Giaccardi's inequality for special case
and inequality introduced by Vasi c and Stankovi c. By using di
erent classes of functions, we formulate families of exponentially
convex functions related to these di erences. We introduce new means
of Cauchy type and prove monotonicity of these means. We, also
exhibit related mean value theorems of Cauchy type. In the fourth
chapter, we consider the nonnegative di erence of Chebyshev's
inequality as Chebyshev functional. We construct symmetric matrices
generated by Chebyshev functional for a class of increasing
functions and prove positive semide niteness of matrices which
implies the exponential and logarithmic convexity of the Chebyshev
functional. Moreover, we demonstrate mean value theorems of Cauchy
type for the Chebyshev functional and its generalized form.
In the last chapter, we start by considering an inequality related
to the Chebyshev's inequality given by A. Lupa s in 1972 but instead
of monotone functions there are convex functions. In addition to
that we consider the reverse of Chebyshev's inequality without
weights introduced by LevinSteckin; here one function is symmetric
increasing and other is continuous convex.By taking the nonnegative
di erences of each inequality, we construct families of
exponentially convex functions. We introduce related Cauchy means
and prove related mean value theorems of Cauchy type. 
