ON EXPONENTIAL TYPE P -FUNCTIONS

In this paper, we introduce and study the concept of exponential type P -function and establish Hermite-Hadamards inequalities for this type of functions. In addition, we obtain some new Hermite-Hadamard type inequalities for functions whose rst derivative in absolute value is exponential type P -function by using Hölder and power-mean integral inequalities. We also extend our initial results to functions of several variables. Next, we point out some applications of our results to give estimates for the approximation error of the integral the function in the trapezoidal formula and for some inequalities related to special means of real numbers. 1. Preliminaries Let : I ! R be a convex function. Then the following inequalities hold r + s 2 1 s r Z s r (u)du (r) + (s) 2 (1) for all r; s 2 I with r < s. Both inequalities hold in the reversed direction if the function is concave. This double inequality is well known as the HermiteHadamard inequality [6]. Note that some of the classical inequalities for means can be derived from Hermite-Hadamard integral inequalities for appropriate particular selections of the mapping . In [5], Dragomir et al. gave the following denition and related Hermite-Hadamard integral inequalities as follow: Denition 1. A nonnegative function : I R ! R is said to be P -function if the inequality ( r + (1 ) s) (r) + (s) 2020 Mathematics Subject Classication. 26A51, 26D10, 26D15. Keywords and phrases. Exponential type convexity, exponential type P -function, HermiteHadamard inequality. selim.numan@giresun.edu.tr; imdat.iscan@giresun.edu.tr, imdati@yahoo.comCorresponding author 0000-0002-5483-6861; 0000-0001-6749-0591. c 2021 Ankara University Communications Facu lty of Sciences University of Ankara-Series A1 Mathematics and Statistics 497 498 S. NUMAN, · I. · IŞCAN holds for all r; s 2 I and 2 (0; 1). Theorem 2. Let 2 P (I), r; s 2 I with r < s and 2 L [r; s]. Then r + s 2 2 s r Z s r (u)du 2 [ (r) + (s)] : (2) Denition 3. [14] Let h : J ! R be a non-negative function, h 6= 0: We say that : I ! R is an h-convex function, or that belongs to the class SX (h; I), if is non-negative and for all u; v 2 I; 2 (0; 1) we have ( r + (1 ) s) h( ) (r) + h(1 ) (s) : If this inequality is reversed, then is said to be h-concave, i.e. 2 SV (h; I). It is clear that, if we choose h( ) = and h( ) = 1, then the h-convexity reduces to convexity and denition of P -function, respectively. Readers can look at [1, 14] for studies on h-convexity. In [11], Kadakal and · Işcan gave the following denition and related HermiteHadamard integral inequalities as follow: Denition 4. A non-negative function : I R ! R is called exponential type convex function if for every r; s 2 I and 2 [0; 1], ( r + (1 ) s) e 1 (r) + e 1 (s): We note that every nonnegative convex function is exponential type convex function. Theorem 5 ( [11]). Let : [r; s] ! R be a exponential type convex function. If r < s and 2 L [r; s], then the following Hermite-Hadamard type inequalities hold: 1 2 [ p e 1] r + s 2 1 s r Z s r (u)du (e 2) [ (r) + (s)] : The main purpose of this paper is to introduce the concept of exponential type P -function which is connected with the concepts of P -function and exponential type convex function and establish some new Hermite-Hadamard type inequality for this class of functions. In recent years many authors have studied error estimations of Hermite-Hadamard type inequalities; for renements, counterparts, generalizations, for some related papers see [2, 3, 4, 5, 7, 8, 9, 10,11,12,13]. 2. The definition of exponential type P -function In this section, we introduce a new concept, which is called exponential type P -function and we give by setting some algebraic properties for the exponential type P -function, as follows: Denition 6. A non-negative function : I R ! R is called exponential type P -function if for every r; s 2 I and 2 [0; 1], ( r + (1 ) s) e + e 2 [ (r) + (s)] : (3) ON EXPONENTIAL TYPE P -FUNCTIONS 499 We will denote by ETP (I) the class of all exponential type P -functions on interval I. We note that, every exponential type P -function is a h-convex function with the function h( ) = e + e 2. Therefore, if ; 2 ETP (I), then i.) + 2 ETP (I) and for c 2 R ( c 0) c 2 ETP (I) (see [14], Proposition 9). ii.) If and g be a similarly ordered functions on I , then : 2 ETP (I) :(see [14], Proposition 10). Also, if : I ! J is a convex and 2 ETP (J) and nondecreasing, then 2 ETP (I) (see [14], Theorem 15). Remark 7. We note that if is satisfy (3), then is a nonnegative function. Indeed, if we rewrite the inequality (3) for = 0 and r = s then 0 (2e 3) (r) for every r 2 I. Thus we have (r) 0 for all r 2 I: Proposition 8. Every exponential type convex function is also a exponential type P -function. Proof. Let : I R! R be an arbitrary exponential type convex function. Then is nonnegative and the following inequality holds ( r + (1 ) s) e 1 (r) + e 1 (s) for every r; s 2 I and 2 [0; 1] : By (r) (r) + (s) and (s) (r) + (s), we obtain desired result. Proposition 9. Every P -function is also a exponential type P -function. Proof. The proof is clear from the following inequalities e 1 and 1 e 1 for all 2 [0; 1] : In this case, we can write 1 e + e 2: Therefore, the desired result is obtained. We can give the following corollary for every nonnegative convex function is also a P -function. Corollary 10. Every nonnegative convex function is also a exponential type P function. Theorem 11. If : [r; s] R ! R is an exponential type P -function, then is bounded on [r; s] : 500 S. NUMAN, · I. · IŞCAN Proof. Let M = max f (r); (s)g : For any x 2 [r; s] ; there exists a 2 [0; 1] such that x = r+(1 ) s. Since is an exponential type P -function on [a; b], we have (x) e + e 2 [ (r) + (s)] 4M(e 1): This shows that is bounded from above. For any x 2 [r; s] ; there exists a 2 [0; 1] such that either x = r+s 2 + or x = r+s 2 : Since it will lose nothing generality we can assume x = r+s 2 + . Thus we can write r + s 2 = 1 2 r + s 2 + + 1 2 r + s 2 2( p e 1) (x) + r + s 2 and from here we have (x) 1 2( p e 1) r + s 2 r + s 2 1 2( p e 1) r + s 2 4M(e 1) = m: This completes the proof. Theorem 12. Let s > r and : [r; s]! R be an arbitrary family of exponential type P -function and let (x) = sup (x). If J = fu 2 [r; s] : (u) <1g is nonempty, then J is an interval and is a exponential type P -function on J . Proof. Let 2 [0; 1] and r; s 2 J be arbitrary. Then ( r + (1 ) s) = sup ( r + (1 ) s) sup e + e 2 [ (r) + (s)] e + e 2 sup (r) + sup (s) = e + e 2 [ (r) + (s)] <1: This shows simultaneously that J is an interval, since it contains every point between any two of its points, and that is an exponential type P -function on J . This completes the proof of theorem. 3. Hermite-Hadamards inequality for exponential type P -functions The goal of this paper is to establish some inequalities of Hermite-Hadamard type for exponential type P -functions. In this section, we will denote by L [r; s] the space of (Lebesgue) integrable functions on [r; s] : ON EXPONENTIAL TYPE P -FUNCTIONS 501 Theorem 13. Let : [r; s] ! R be a exponential type P -function. If r < s and 2 L [r; s], then the following Hermite-Hadamard type inequalities hold: 1 4 [ p e 1] r + s 2 1 s r Z s r (u)du (2e 4) [ (r) + (s)] : (4) Proof. Since is a exponential type P -function, we get r + s 2 = 1 2 [ r + (1 ) s] + 1 2 [ s+ (1 ) r] 2 p e 1 [ ( r + (1 ) s) + ( s+ (1 ) r)] : By taking integral in the last inequality with respect to 2 [0; 1], we deduce that r + s 2 4 s r p e 1 Z s


Preliminaries
Let : I ! R be a convex function. Then the following inequalities hold r + s 2 1 s r Z s r (u)du (r) + (s) 2 (1) for all r; s 2 I with r < s. Both inequalities hold in the reversed direction if the function is concave. This double inequality is well known as the Hermite-Hadamard inequality [6]. Note that some of the classical inequalities for means can be derived from Hermite-Hadamard integral inequalities for appropriate particular selections of the mapping .
In [5], Dragomir  (2) De…nition 3. [14] Let h : J ! R be a non-negative function, h 6 = 0: We say that : I ! R is an h-convex function, or that belongs to the class SX (h; I), if is non-negative and for all u; v 2 I; 2 (0; 1) we have If this inequality is reversed, then is said to be h-concave, i.e. 2 SV (h; I). It is clear that, if we choose h( ) = and h( ) = 1, then the h-convexity reduces to convexity and de…nition of P -function, respectively.
Readers can look at [1,14] for studies on h-convexity. In [11], Kadakal and · Işcan gave the following de…nition and related Hermite-Hadamard integral inequalities as follow:

De…nition 4. A non-negative function
: I R ! R is called exponential type convex function if for every r; s 2 I and 2 [0; 1], We note that every nonnegative convex function is exponential type convex function. The main purpose of this paper is to introduce the concept of exponential type P -function which is connected with the concepts of P -function and exponential type convex function and establish some new Hermite-Hadamard type inequality for this class of functions. In recent years many authors have studied error estimations of Hermite-Hadamard type inequalities; for re…nements, counterparts, generalizations, for some related papers see [2,3,4,5,7,8,9,10,11,12,13].

The definition of exponential type P -function
In this section, we introduce a new concept, which is called exponential type P -function and we give by setting some algebraic properties for the exponential type P -function, as follows: De…nition 6. A non-negative function : I R ! R is called exponential type P -function if for every r; s 2 I and 2 [0; 1], ON EXPONENTIAL TYPE P -FUNCTIONS 499 We will denote by ET P (I) the class of all exponential type P -functions on interval I.
We note that, every exponential type P -function is a h-convex function with the function h( ) = e + e 1 2. Therefore, if ; 2 ET P (I), then i.) + 2 ET P (I) and for c 2 R ( c 0) c 2 ET P (I) (see [14], Proposition 9).
ii.) If and g be a similarly ordered functions on I , then : 2 ET P (I) :(see [14], Proposition 10).

Remark 7.
We note that if is satisfy (3), then is a nonnegative function. Indeed, if we rewrite the inequality (3) for = 0 and r = s then for every r 2 I. Thus we have (r) 0 for all r 2 I: Proposition 8. Every exponential type convex function is also a exponential type P -function.
Proof. Let : I R ! R be an arbitrary exponential type convex function. Then is nonnegative and the following inequality holds for every r; s 2 I and 2 [0; 1] : By (r) (r) + (s) and (s) (r) + (s), we obtain desired result.
Proposition 9. Every P -function is also a exponential type P -function.
Proof. The proof is clear from the following inequalities Therefore, the desired result is obtained.
We can give the following corollary for every nonnegative convex function is also a P -function.
Corollary 10. Every nonnegative convex function is also a exponential type Pfunction. : Since it will lose nothing generality we can assume x = r+s 2 + . Thus we can write r + s 2 = 1 2 and from here we have This completes the proof.
Theorem 12. Let s > r and : [r; s] ! R be an arbitrary family of exponential type P -function and let (x) = sup (x). If J = fu 2 [r; s] : (u) < 1g is nonempty, then J is an interval and is a exponential type P -function on J.
Proof. Let 2 [0; 1] and r; s 2 J be arbitrary. Then This shows simultaneously that J is an interval, since it contains every point between any two of its points, and that is an exponential type P -function on J. This completes the proof of theorem.

Hermite-Hadamard's inequality for exponential type P -functions
The goal of this paper is to establish some inequalities of Hermite-Hadamard type for exponential type P -functions. In this section, we will denote by L [r; s] the space of (Lebesgue) integrable functions on [r; s] : Theorem 13. Let : [r; s] ! R be a exponential type P -function. If r < s and 2 L [r; s], then the following Hermite-Hadamard type inequalities hold: Proof. Since is a exponential type P -function, we get This completes the proof.

Some new inequalities for exponential type P -functions
The main purpose of this section is to establish new estimates that re…ne Hermite-Hadamard inequality for functions whose …rst derivative in absolute value is exponential type P -function. Dragomir and Agarwal [4] used the following lemma: (5) where A is the arithmetic mean.
Proof. Using Lemma 15 and the inequality jf 0 (tr + (1 t)s)j e t + e 1 t 2 [jf 0 (r)j + jf 0 (s)j] ; we get This completes the proof of theorem.

ON EXPONENTIAL TYPE P -FUNCTIONS 503
Theorem 17. Let f : I ! R be a di¤ erentiable mapping on I , r; s 2 I with r < s and assume that f 0 2 L [r; s]. If jf 0 j q ; q > 1; is an exponential type P -function on interval [r; s], then the following inequality holds where 1 p + 1 q = 1 and A is the arithmetic mean. Proof. Using Lemma 15, Hölder's integral inequality and the following inequality which is the exponential type P -function of jf 0 j q , we get This completes the proof of theorem.
Theorem 18. Let f : I R ! R be a di¤ erentiable mapping on I , r; s 2 I with r < s and assume that f 0 2 L [r; s]. If jf 0 j q ; q 1; is an exponential type P -function on the interval [r; s], then the following inequality holds s r Proof. From Lemma 15, well known power-mean integral inequality and the property of exponential type P -function of jf 0 j q , we obtain This completes the proof of theorem.
Corollary 19. Under the assumption of Theorem 18, If we take q = 1 in the inequality (7), then we get the following inequality: This inequality coincides with the inequality (5).

An extention of Theorem 16
In this section we will denote by K an open and convex set of R n (n 1).
We say that a function f : for all x; y 2 K and t 2 [0; 1].
Using the above lemma we will prove an extension of Theorem 16 to functions of several variables.
Proposition 21. Assume f : K R n ! R + is a exponential type P -function on K. Then for any x; y 2 K and any u; v 2 (0; 1) with u < v the following inequality holds 1 2 (v u) 8 p e 2e 7 A (f (ux + (1 u)y) ; f (vx + (1 v)y)) : Proof. We …x x; y 2 K and u; v 2 (0; 1) with u < v. Since f is exponential type P -function, by Lemma 20 it follows that the function and we deduce that relation (8) holds true.
Remark 22. We point out that a similar result as those of Proposition 21 can be stated by using Theorem 17 and Theorem 18 .

Applications to the trapezoidal formula
Assume } is a division of the interval [r; s] such that } : r = x 0 < x 1 < ::: < x n 1 < x n = s: For a given function f : [r; s] ! R we consider the trapezoidal formula It is well known that if f is twice di¤erentiable on (r; s) and M = sup x2(r;s) jf 00 (x)j < 1 then Z s r f (x)dx = T (f; }) + E (f; }) ;