Application of Functionals in Creating Inequalities

In our research, we apply the theory of positive linear functionals to convex analysis. Let us remember the initial notions related to positive linear functionals on the space of real functions. Let S be a nonempty set, and let F = F(S) be a subspace of the linear space of all real functions on domain S. We assume that space F contains unit function u defined by u(s) = 1 for every s ∈ S. Such space F contains every real constant κwithin the meaning of κ = κu and every composite function f(g) of a function g ∈ F and an affine function f : R → R. Actually, if f(x) = κ1x + κ2, then the composition f (g) = κ1g + κ2u (1) belongs to F . Let L = L(F) be the space of all linear functionals on space F . Functional L ∈ L is said to be unital (normalized) if L(u) = 1. Such functional has property L(κu) = κ for every real constant κ. If g ∈ F is a function and if L ∈ L is a unital functional, then affine function f : R → R satisfies equality f (L (g)) = L (f (g)) . (2) Functional L ∈ L is said to be positive (nonnegative) if inequality L(g) ≥ 0 holds for every nonnegative function g ∈ F . If a pair of functions g1, g2 ∈ F satisfies inequality g1(s) ≤ g2(s) for every s ∈ S, then it follows that L (g1) ≤ L (g2) . (3) If g ∈ F is a function with the image in interval [a, b] (i.e., au ≤ g ≤ bu), then every positive unital functional L ∈ L meets inclusion L(g) ∈ [a, b] (i.e., a ≤ L(g) ≤ b). The same is true for each closed interval I ⊆ R. Introducing a continuous convex function, we can expose the functional form of Jensen’s inequality. Theorem A. Let g ∈ F be a function with the image in closed interval I ⊆ R, and let L ∈ L be a positive unital functional. Then each continuous convex functionf : I → R such that f(g) ∈ F satisfies inequality f (L (g)) ≤ L (f (g)) . (4)


Introduction
In our research, we apply the theory of positive linear functionals to convex analysis.Let us remember the initial notions related to positive linear functionals on the space of real functions.
Let  be a nonempty set, and let F = F() be a subspace of the linear space of all real functions on domain .We assume that space F contains unit function  defined by () = 1 for every  ∈ .Such space F contains every real constant  within the meaning of  =  and every composite function () of a function  ∈ F and an affine function  : R → R. Actually, if () =  1  +  2 , then the composition belongs to F. Let L = L(F) be the space of all linear functionals on space F. Functional  ∈ L is said to be unital (normalized) if () = 1.Such functional has property () =  for every real constant .If  ∈ F is a function and if  ∈ L is a unital functional, then affine function  : R → R satisfies equality  ( ()) =  ( ()) . ( Functional  ∈ L is said to be positive (nonnegative) if inequality () ≥ 0 holds for every nonnegative function  ∈ F. If a pair of functions  1 ,  2 ∈ F satisfies inequality  1 () ≤  2 () for every  ∈ , then it follows that  ( 1 ) ≤  ( 2 ) . ( If  ∈ F is a function with the image in interval [, ] (i.e.,  ≤  ≤ ), then every positive unital functional  ∈ L meets inclusion () ∈ [, ] (i.e.,  ≤ () ≤ ).The same is true for each closed interval  ⊆ R.
Introducing a continuous convex function, we can expose the functional form of Jensen's inequality.
Theorem A. Let  ∈ F be a function with the image in closed interval  ⊆ R, and let  ∈ L be a positive unital functional.
Then each continuous convex function  :  → R such that () ∈ F satisfies inequality We will consider convex functions in bounded closed interval [, ] with endpoints  < .Each point  ∈ [, ] can be represented by the unique binomial convex combination where Convex function  : [, ] → R is bounded by two lines.The secant line of function  passes through graph points (, ()) and (, ()), and its equation is Let  ∈ (, ) be an interior point.The support lines of function  pass through graph point (, ()).Each support line is specified by slope coefficient  ∈ [  (−),   (+)], and its equation is The support-secant line inequality holds for every  ∈ [, ].
In 1931, Jessen (see [1,2]) stated the functional form of Jensen's inequality for convex functions in interval  ⊆ R. In 1988, I. Rasa and I. Ras ¸a (see [3]) pointed out that  must be closed otherwise it could happen that () ∉  and that  must be continuous otherwise it could happen that the inequality in formula (4) does not apply.In Theorem A, we have taken into account I. Rasa and I. Ras ¸a's remarks.Some generalizations of the functional form of Jensen's inequality can be found in [4].
A concise book on functional analysis, which contains an essential overview of operator theory and indicates the importance of positive linear functionals, is certainly the book in [5].

Main Results
We firstly present the extension of the inequality in formula (4) where the first term If  ∈ {, }, we rely on the continuity of  using a support line at a point of open interval (, ) that is close enough to .Given  > 0, we can find  ∈ (, ) so that  () −  <  sup  () .
Formula ( 10) can be expressed in the form which includes the convex combination of interval endpoints  and .The respective form of Lemma 1 is as follows.

Corollary 2. Let 𝑔 ∈ F be a function with the image in [𝑎, 𝑏],
and let  ∈ L be a positive unital functional.Let Then each continuous convex function  : Proof.As regards to the last terms of formulae ( 10) and ( 17), we have because of the affinity of  sec  and its coincidence with  at endpoints.
In order to refine the inequality in formula (10), we will combine the secant lines of convex function  with positive unital functionals.Lemma 3. Let  ∈ (, ) be a point.
Suppose that function  ∈ F has the image in [, ] and is not identically equal to  or .Such function satisfies inequality ( 1 ) ≤  ≤ ( 2 ) for some number  ∈ (, ) and some pair of points  1 ,  2 ∈ .In that case, we can find a pair of functionals  1 ,  2 ∈ L meeting related inequality For example, we can take the point evaluations at  1 and  2 , that is, the functionals defined by  1 () = ( 1 ) and  2 () = ( 2 ) for every function  ∈ F.
In the main theorem, we use functionals  1 and  2 satisfying the inequality in formula (20).Then each continuous convex function  : [, ] → R such that () ∈ F satisfies the series of inequalities Proof.By applying the convexity of  to convex combination By applying the left-hand side of formula (10) to  1 and  2 , we obtain As the right-hand side of formula (19) with  = (), the inequality holds for every  ∈ .By acting with  1 in the above inequality and using assumption  1 () ∈ [, ], we find and similarly, by acting with  2 and using the assumption  2 () ∈ [, ], we find Multiplication by  1 and  2 and then summation yield Using main secant  sec  , we reach conclusion Putting together the inequalities in formulae ( 22), ( 23), (27), and (28) into a series, we achieve the inequality in formula (21).The geometric presentation of the series of inequalities in formula ( 21) is created in Figure 1.The inequality terms are represented by five black dots above point ().
To emphasize interval endpoints  and , we present the following version of Theorem 4. where Proof.To calculate coefficients , , and , we include convex combinations  1 () =  1  +  1  and  2 () =  2  +  2 .Then the fourth term of formula (21) takes the form Taking the coefficient of () and using formula (6), we calculate Similarly we determine  and .
Let us finish the section by presenting the generalization of Theorem 4 that uses several secant lines.(33)

Applications to Integral and Discrete Inequalities
We firstly utilize Lemma 1 to obtain a very general integral inequality.for every  ∈ F is positive and unital.The first term of formula ( 34) is equal to (()), the second term is equal to (()), and the third term is equal to  sec  (()).Thus, formula (34) fits into the frame of formula (10), and it is valid for a continuous convex function .
Let us verify that the inequality in formula (34) applies to a convex function which is not continuous at endpoints.We observe the position of point If  ∈ (, ), then we may utilize continuous extension f of /(, ) to [, ] in formula (34).The first two terms are the same as we use , and the last terms satisfy inequality Respecting all considerations, we may conclude that the inequality in formula (34) applies to any convex function .

The inequality in formula (34) is the extended version of Jensen's inequality for the ratio of integrals in interval [𝑎, 𝑏],
as well as the generalized form of the Fejér and Hermite-Hadamard inequality.
Let us demonstrate the simplifications of the inequality in formula (34) relating to the identity, unit, and symmetric function.
and functional convex combination  =  1  1 +  2  2 , we get By further calculating the functional terms according to formula (21), we obtain the integral terms of formula (44).
The series of inequalities in formula (44) with () =  and ℎ() = 1 gives the refinement of the Hermite-Hadamard inequality in formula (43) as follows: The above refinement holds for each  ∈ (, ).The version of the above refinement was obtained in [10] acting to every  ∈ F, and coefficients Functionals  1 and  2 are certainly positive and unital.
The inequality in formula (49) is the extension and refinement of the famous discrete form of Jensen's inequality (see [11]).

concerning interval [𝑎, 𝑏]. Lemma 1. Let 𝑔 ∈ F be a function with the image in [𝑎, 𝑏], and let 𝐿 ∈ L be a positive unital functional.
by using the Jensen type inequality for convex combinations with the common center.That inequality was used to refine some important means.The inequality in formula (44) with identity function () =  and a symmetric function satisfying equation ℎ() = ℎ( +  − ) provides the refinements of the Fejér inequality in formula (40).At the end, let us present the discrete version of Corollary 8. Point evaluations (  ) and ℎ(  ) will be shortened by   and ℎ  , respectively.Let  ∈ (, ) be a point.Let  : [, ] → R be a function such that () ∈ [, ] for  ∈ [, ] and () ∈ [, ] for  ∈ [, ], and let ℎ : [, ] → R be a positive function.Let  1 , . . .,   ∈ [, ] and  +1 , . . .,   ∈ [, ] be points.Then each convex function  : [, ] → R satisfies the series of inequalities Proof.Let F be the space of all real functions on domain  = [, ].We can apply the proof of Corollary 8 to summarizing linear functionals