Deciding where to
begin is a major step. One procedure is to lay out all necessary
preliminary material, introduce the major ideas in their most
general setting, prove the theorems and then specialize to obtain
classical results and various applications. We experience convexity
all the times and in many ways. The most prosaic example is our
upright position, which is secured as long as the vertical
projection of our center of gravity lies inside the convex envelope
of our feet. Also convexity has a great impact on our every day life
through numerous applications in industry, business, medicine and
art. So do the problems of optimum allocation of resources and
equilibrium of non cooperative games. The theory of convex functions
is a part of the general subject of convexity, since a convex
function is one whose epigraph is a convex set. Nonetheless it is an
important theory, which touches almost all branches of mathematics.
In calculus, the mean value theorem states, roughly, that given a
section of a smooth curve, there is a point on that section at which
the derivative (slope) of the curve is equal (parallel) to the
”average” derivative of the section. It is used to prove theorems
that make global conclusions about a function on an interval
starting from local hypotheses about derivatives at points of the
interval. This theorem can be understood concretely by applying it
to motion: if a car travels one hundred miles in one hour, so that
its average speed during that time was 100 miles per hour, then at
some time its instantaneous speed must have been exactly 100 miles
An early version of this theorem was first described by Parameshvara
(1370-1460) from the Kerala school of astronomy and mathematics in
his commentaries on Govindasvami and Bhaskara II. The mean value
theorem in its modern form was later stated by Augustin Louis Cauchy
(1789-1857). It is one of the most important results in differential
calculus, as well as one of the most important theorems in
mathematical analysis, and is essential in proving the fundamental
theorem of calculus. The mean value theorem can be used to prove
Taylor’s theorem, of which it is a special case. We use this Mean
value theorem and its other generalized version to define new
In the first chapter some basic notions and results from the theory
of means and convex functions are being introduced along with
classical results of convex functions.
In the second chapter we define some further results about
logarithmic convexity of differences of of power means for positive
linear functionals as well as some related results.
In the third chapter we define new means of Cauchy’s type. We prove
that this mean is monotonic. Also we give some applications of this
In the fourth chapter we give Cauchy’s means of Boas type for non
positive measure. We show that these Cauchy’s means are monotonic.
In the fifth chapter, we give definition of Cauchy means of Mercer’s
type. Also, we show that these means are monotonic.
In the sixth chapter, we define the generalization of results given
by S. Simi´c, for logconvexity for differences of mixed symmetric
means. We also present related Cauchy’s means.
In the last chapter we give an improvement and reversion of well
known Ky-Fan inequality. Also we introduce in this chapter Levinson
means of Cauchy’s type. We prove that these means are monotonic.