Abstract
This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided.
Similar content being viewed by others
1 Introduction and preliminaries
The concept of convexity has played a crucial role in the advancement of the theory of inequalities.
Definition 1
Let \(F \subseteq \mathbb{R}\) be an interval. Then, a function \(\Lambda :F \rightarrow \mathbb{R}\) is said to be convex if
holds for all \(\vartheta ,\varpi \in F \) and \(\mu \in {}[ 0,1]\).
If inequality (1.1) holds in the reverse direction, then Λ is said to be concave on interval \(F \neq \emptyset \).
Numerous generally accepted results from the theory of inequalities can be obtained using the properties of functions; see [2, 8, 11, 14–17, 21, 23, 28, 33] and the references therein.
The Hermite–Hadamard (H–H) inequality is a well-explored and renowned result concerning convex functions, stating that if \(\Lambda :F \rightarrow \mathbb{R}\) is a convex function in F for all \(\vartheta ,\varpi \in F \) with \(\vartheta <\varpi \), then
Interested readers may refer to the monographs [1, 6, 7, 10, 12, 20, 24–27, 32, 34, 35].
Definition 2
[29] Let \(F \subset \mathbb{R} \) be an interval and k be a positive number. A function \(\Lambda :F \subset \mathbb{R} \rightarrow \mathbb{R}\) is called strongly convex with modulus k if
for all \(\vartheta ,\varpi \in F \) and \(\mu \in {}[ 0,1]\).
It is clear that every strongly convex function is also a convex function.
Theorem 1
If a function \(\Lambda :F \rightarrow \mathbb{R} \) is a strongly convex function with modulus k, then
for all \(\vartheta ,\varpi \in F \) with \(\vartheta <\varpi \).
Definition 3
[31] Let \(h:J\rightarrow \mathbb{R} \) be a nonnegative function, \(h\neq 0\). A function \(\Lambda :F \rightarrow \mathbb{R} \) is called h-convex, if Λ is nonnegative, and for all \(\vartheta ,\varpi \in F \), \(\mu \in ( 0,1 ) \), we have
If this inequality is reversed, then Λ is said to be h-concave. Clearly, if we substitute \(h(\mu )=\mu \), then the h-convex functions reduce to the classical convex functions; see [5].
Definition 4
[3] Let \(( \chi , \Vert \cdot \Vert ) \) be a real normed space, and ℵ stands for a convex subset of χ, \(h: ( 0,1 ) \rightarrow ( 0,\infty ) \) is a given function, and k is a positive constant. A function \(\Lambda :\aleph \rightarrow \mathbb{R} \) is called strongly h-convex with module k if
for all \(\vartheta ,\varpi \in \aleph \) and \(\mu \in ( 0,1 ) \).
Theorem 2
[22] Let \(h: ( 0,1 ) \rightarrow ( 0,\infty ) \) be a given function. If a function \(\Lambda :F \subseteq \mathbb{R} \rightarrow \mathbb{R} \) is Lebesgue integrable and strongly h-convex with module \(k>0\), then
for all \(\vartheta ,\varpi \in F \) with \(\vartheta <\varpi \).
In [30], Tekin et al. gave the following definition and related H–H-type inequality as follows:
Definition 5
Let \(n\in \mathbb{N} \). A nonnegative function \(\Lambda :F \subset \mathbb{R}\rightarrow \mathbb{R}\) is called n-polynomial convex if for every \(\vartheta ,\varpi \in F \) and \(\mu \in [ 0,1 ] \),
Theorem 3
Let \(\Psi : [ \vartheta ,\varpi ] \rightarrow \mathbb{R} \) be an n-polynomial convex function. If \(a< b\) and \(\Psi \in L [ \vartheta ,\varpi ] \), then
In [4], Ataman et al. introduced the class of strongly n-polynomial convex functions and related H–H-type inequality as follows:
Definition 6
Let \(n\in \mathbb{N} \), \(F \subset \mathbb{R}\) be an interval and k be a positive number. A nonnegative function \(\Lambda :F \subset \mathbb{R}\rightarrow \mathbb{R}\) is called strongly n-polynomial convex with modulus k if for every \(\vartheta ,\varpi \in F \) and \(\mu \in [ 0,1 ] \),
Theorem 4
Let \(\Psi : [ \vartheta ,\varpi ] \rightarrow \mathbb{R} \) be a strongly n-polynomial convex function with modulus k. If \(\vartheta <\varpi \) and \(\Psi \in L [ \vartheta ,\varpi ] \), then
In [18], Kadakal et al. introduced the class of generalized n-polynomial convex function and related H–H-type inequality as follows:
Definition 7
Let \(n\in \mathbb{N} \) and \(\alpha _{j}\geq 0\) (\(j=\overline{1,n} \)) such that \(\sum_{j=1}^{n}\alpha _{j}>0\). A nonnegative function \(\Lambda :F \subset \mathbb{R}\rightarrow \mathbb{R}\) is called generalized n-polynomial convex function if for every \(\vartheta ,\varpi \in F \) and \(\mu \in [ 0,1 ] \),
Theorem 5
Let \(\Lambda : [ \vartheta ,\varpi ] \rightarrow \mathbb{R} \) be a generalized n-polynomial convex function. If \(\vartheta <\varpi \) and \(\Lambda \in L [ \vartheta ,\varpi ] \), then
Theorem 6
(Hölder-İşcan integral inequality [13])
Let \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1\). If Λ and Θ are real functions defined on \([ \vartheta ,\varpi ] \) and if \(\vert \Lambda \vert ^{p}\), \(\vert \Theta \vert ^{q}\) are integrable functions on \([ \vartheta ,\varpi ] \), then
Theorem 7
(Improved power-mean integral inequality [19])
Let \(q\geq 1\). If Λ and Θ are real functions defined on \([ \vartheta ,\varpi ] \) and \(\vert \Lambda \vert \), \(\vert \Lambda \vert \vert \Theta \vert ^{q}\) are integrable functions on \([ \vartheta ,\varpi ] \), then
2 The definition of generalized strongly n-polynomial convex functions
In this section, we will add a new definition called a generalized strongly n-polynomial convex function and its basic algebraic properties.
Definition 8
Let \(n\in \mathbb{N} \), \(\alpha _{j}\geq 0\) (\(j=\overline{1,n} \)) such that \(\sum_{j=1}^{n}\alpha _{j}>0\), \(F \subset \mathbb{R}\) be an interval, and k be a positive number. A nonnegative function \(\Lambda :F \subset \mathbb{R}\rightarrow \mathbb{R}\) is called the generalized strongly n-polynomial convex function \(( \mathit{GSPOLC} ) \) if for every \(\vartheta ,\varpi \in F \) and \(\mu \in [ 0,1 ] \),
We will denote by \(\mathit{GSPOLC} ( F ) \) the class of all generalized strongly n-polynomial convex functions on F.
Note that every GSPOLC is a strongly h-convex function with the function
Therefore, if \(\Lambda ,\Theta \in \mathit{GSPOLC} ( F ) \), it can be easily seen that \(\Lambda +\Theta \in \mathit{GSPOLC} ( F ) \) and for \(b\in \mathbb{R} \) (\(b\geq 0\)), \(b\Lambda \in \mathit{GSPOLC} ( F ) \).
Also, if \(\Lambda :F \rightarrow J\) is convex and \(\Theta \in \mathit{GSPOLC} ( J ) \) and nondecreasing, then \(\Theta \circ \Lambda \in \mathit{GSPOLC} ( F ) \) (see [31], Theorem 15).
Remark 1
For \(n=1\) and \(k=0\), Definition 8 reduces to Definition 1.
Remark 2
For \(\alpha _{j}=1\) (\(j=\overline{1,n} \)) and \(k=0\), Definition 8 reduces to Definition 5.
Remark 3
For \(k=0\), Definition 8 reduces to Definition 7.
Remark 4
For \(n=1\), Definition 8 reduces to Definition 2.
Remark 5
For \(\alpha _{j}=1\) (\(j=\overline{1,n} \)), Definition 8 reduces to Definition 6.
Remark 6
Every nonnegative strongly convex function is GSPOLC. It is obvious from the inequalities
for all \(\mu \in [ 0,1 ] \) and \(n\in \mathbb{N} \).
Example 1
Let \(\Lambda ( {\sigma } ) ={\sigma }^{2}\) and \([ \vartheta ,\varpi ] = [ -1,1 ] \). Then, Λ is GSPOLC with modulus \(k=1\).
3 H–H inequality for generalized strongly n-polynomial convex functions
This section aims to establish the H–H inequality for the new class of functions.
Theorem 8
Let \(\Lambda : [ \vartheta ,\varpi ] \rightarrow \mathbb{R} \) be GSPOLC with modulus k. If \(\vartheta <\varpi \) and \(\Lambda \in L [ \vartheta ,\varpi ] \), then
Proof
Using the generalized strongly n-polynomial convexity of Λ, one has
Now integrating the above inequality with respect to \(\mu \in [ 0,1 ] \), one gets
which completes the left-hand side of inequality (3.1). For the right-hand side of inequality (3.1), changing the variable of integration as \({\sigma }=\mu \vartheta +(1-\mu )\varpi \) and using generalized strongly n-polynomial convexity of Λ, one obtains
where
□
Remark 7
For \(n=1\) and \(k=0\), inequality (3.1) reduces to inequality (1.2).
Remark 8
For \(\alpha _{j}=1\) (\(j=\overline{1,n} \)) and \(k=0\), inequality (3.1) reduces to inequality (1.9).
Remark 9
For \(k=0\), inequality (3.1) reduces to inequality (1.13).
Remark 10
For \(n=1\), inequality (3.1) reduces to inequality (1.4).
Remark 11
For \(\alpha _{j}=1\) (\(j=\overline{1,n} \)), inequality (3.1) reduces to inequality (1.11).
4 Refinements of H–H inequality
Let us recall the following crucial lemma that we will use in the future:
Lemma 1
Let \(\Lambda :F ^{\circ }\rightarrow \mathbb{R} \) be a differentiable mapping on \(F ^{\circ }\), \(\vartheta ,\varpi \in F ^{\circ }\) with \(\vartheta <\varpi \). If \(\Lambda ^{\prime }\in L [ \vartheta ,\varpi ] \), then
Theorem 9
Let \(\Lambda :F \rightarrow \mathbb{R} \) be a differentiable function on \(F ^{\circ }\), \(\vartheta ,\varpi \in F ^{\circ }\) with \(\vartheta <\varpi \), and let \(\Lambda ^{\prime }\in L [ \vartheta ,\varpi ] \). If \(\vert \Lambda ^{\prime } \vert \) is GSPOLC with modulus k on \([ \vartheta ,\varpi ] \), then the following inequality holds for \(\mu \in [ 0,1 ] \):
where A is the arithmetic mean.
Proof
Using Lemma 1 and the inequality
one obtains
where
□
Corollary 1
If one takes \(n=1\) and \(k=0\) in (4.1), then
Inequality (4.2) coincides with the inequality in [9, Theorem 2.2].
Corollary 2
If one takes \(\alpha _{j}=1\) (\(j=\overline{1,n} \)) and \(k=0\) in (4.1), then
Inequality (4.3) coincides with the inequality in [30, Theorem 5].
Corollary 3
If one takes \(k=0\) in (4.1), then
Inequality (4.4) coincides with the inequality in [18, Theorem 5].
Corollary 4
If one takes \(n=1\) in (4.1), then
Corollary 5
If one takes \(\alpha _{j}=1\) (\(j=\overline{1,n} \)) in the inequality, then
Inequality (4.5) coincides with the inequality in [4, Theorem 5].
Theorem 10
Let \(\Lambda :F \rightarrow \mathbb{R} \) be a differentiable function on \(F ^{\circ }\), \(\vartheta ,\varpi \in F ^{\circ }\) with \(\vartheta <\varpi \), \(q>1\), \(\frac{1}{p}+\frac{1}{q}=1\), and assume that \(\Lambda ^{\prime }\in L [ \vartheta ,\varpi ] \). If \(\vert \Lambda ^{\prime } \vert ^{q}\) is GSPOLC with modulus k on \([ \vartheta ,\varpi ] \), then
where A is the arithmetic mean.
Proof
Using Lemma 1, the Hölder integral inequality and the inequality
one gets
where
□
Corollary 6
If one takes \(n=1\) and \(k=0\) in (4.6), then
Inequality (4.7) coincides with the inequality in [9, Theorem 2.3].
Corollary 7
If one takes \(\alpha _{j}=1\) (\(j=\overline{1,n} \)) and \(k=0\) in (4.6), then
Inequality (4.8) coincides with the inequality in [30, Theorem 6].
Corollary 8
If one takes \(k=0\) in (4.6), then
Inequality (4.9) coincides with the inequality in [18, Theorem 6].
Corollary 9
If one takes \(n=1\) in (4.6), then
Corollary 10
If one takes \(\alpha _{j}=1\) (\(j=\overline{1,n} \)) in (4.6), then
Inequality (4.10) coincides with the inequality in [4, Theorem 5].
Theorem 11
Let \(\Lambda :F \subseteq \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function on \(F ^{\circ }\), \(\vartheta ,\varpi \in F ^{\circ }\) with \(\vartheta <\varpi \), \(q\geq 1\), and assume that \(\Lambda ^{\prime }\in L [ \vartheta ,\varpi ] \). If \(\vert \Lambda ^{\prime } \vert ^{q}\) is GSPOLC with modulus k on \([ \vartheta ,\varpi ] \), then
where A is the arithmetic mean.
Proof
From Lemma 1, the Hölder integral inequality, and the property of GSPOLC of \(\vert \Lambda ^{\prime } \vert ^{q}\), one obtains
where
□
Corollary 11
Under the assumption of Theorem 11with \(q=1\), one gets the conclusion of Theorem 9.
Corollary 12
If one takes \(n=1\), \(k=0\) and \(q=1\) in (4.11), then
Inequality (4.12) coincides with the inequality in [9, Theorem 1].
Corollary 13
If one takes \(\alpha _{j}=1\) (\(j=\overline{1,n} \)) and \(k=0\) in (4.11), then
Inequality (4.13) coincides with the inequality in [30, Theorem 1]. Also, if one takes \(q=1\) in (4.13), then one gets
Inequality (4.14) coincides with the inequality in [30, Theorem 1].
Corollary 14
If one takes \(n=1\) in (4.11), then
Also, if one takes \(q=1\) in (4.15), then
Theorem 12
Let \(\Lambda :F \rightarrow \mathbb{R} \) be a differentiable function on \(F ^{\circ }\), \(\vartheta ,\varpi \in F ^{\circ }\) with \(\vartheta <\varpi \), \(q>1\), \(\frac{1}{p}+\frac{1}{q}=1\), and assume that \(\Lambda ^{\prime }\in L [ \vartheta ,\varpi ] \). If \(\vert \Lambda ^{\prime } \vert ^{q}\) is GSPOLC with modulus k on \([ \vartheta ,\varpi ] \), then
Proof
Using Lemma 1, the Hölder-İşcan integral inequality, and the property of GSPOLC of \(\vert \Lambda ^{\prime } \vert ^{q}\), one has
where
□
Corollary 15
If one takes \(n=1\) and \(k=0\) in (4.16), then
Inequality (4.17) coincides with the inequality in [13, Theorem 3.2].
Corollary 16
If one takes \(\alpha _{j}=1\) (\(j=\overline{1,n} \)) and \(k=0\) in (4.16), then
Inequality (4.18) coincides with the inequality in [30, Theorem 3.2].
Corollary 17
If one takes \(k=0\) in (12), then
Corollary 18
If one takes \(n=1\) in (4.16), then
Corollary 19
If one take \(\alpha _{j}=1\) (\(j=\overline{1,n} \)) in (4.16), then
Remark 12
Inequality (4.16) gives better results than inequality (4.6). Let us show that
From the concavity of the function \(h: [ 0,\infty ) \rightarrow \mathbb{R} \), \(h({\sigma })={\sigma }^{r}\), \(0< r\leq 1\), one gets
This is the desired result.
Theorem 13
Let \(\Lambda :F \subseteq \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function on \(F ^{\circ }\), \(\vartheta ,\varpi \in F ^{\circ }\) with \(\vartheta <\varpi \), \(q\geq 1\), and assume that \(\Lambda ^{\prime }\in L [ \vartheta ,\varpi ] \). If \(\vert \Lambda ^{\prime } \vert ^{q}\) is GSPOLC with modulus k on the \([ \vartheta ,\varpi ] \), then
where
Proof
From Lemma 1, the improved power-mean integral inequality, and the property of GSPOLC of \(\vert \Lambda ^{\prime } \vert ^{q}\), one obtains
where
□
Corollary 20
If one takes \(n=1\) in (4.19), then
Corollary 21
If one takes \(n=1\) and \(k=0\) in (4.19), then
Also, if one takes \(q=1\) in (4.19), then
Corollary 22
If one takes \(a_{j}=1\) (\(j=\overline{1,n} \)) and \(k=0\) in (4.19), then
Inequality (4.20) coincides with the inequality in [30, Theorem 9].
Corollary 23
If one takes \(a_{j}=1\) (\(j=\overline{1,n} \)) in (4.19), then
Inequality (4.21) coincides with the inequality in [4, Theorem 9].
Remark 13
Inequality (4.19) gives better result than the inequality (4.11).
Using concavity of the function \(h: [ 0,\infty ) \rightarrow \mathbb{R} \), \(h({\sigma })={\sigma }^{\lambda }\), \(0<\lambda \leq 1\), one gets
where
This completes the proof.
5 Applications for special means
Throughout this section, the following notations will be used for special means of two nonnegative numbers ϑ, ϖ with \(\varpi >\vartheta \):
-
1.
The arithmetic mean
$$ A(\vartheta ,\varpi )=\frac{\vartheta +\varpi }{2},\quad \vartheta , \varpi \geq 0. $$ -
2.
The geometric mean
$$ G(\vartheta ,\varpi )=\sqrt{\vartheta \varpi },\quad \vartheta , \varpi \geq 0. $$ -
3.
The harmonic mean
$$ H(\vartheta ,\varpi )=\frac{2\vartheta \varpi }{\vartheta +\varpi },\quad \vartheta ,\varpi >0. $$ -
4.
The logarithmic mean
$$ L(\vartheta ,\varpi )=\textstyle\begin{cases} \frac{\varpi -\vartheta }{\ln \varpi -\ln \vartheta }, & \vartheta \neq \varpi, \\ \vartheta , & \vartheta =\varpi; \end{cases}\displaystyle \quad \vartheta ,\varpi >0. $$ -
5.
The p-logarithmic mean
$$ L_{p}(\vartheta ,\varpi )=\textstyle\begin{cases} ( \frac{\varpi ^{p+1}-\vartheta ^{p+1}}{(p+1)(\varpi -\vartheta )} ) ^{\frac{1}{p}}, & \vartheta \neq \varpi ,p\in \mathbb{R} \backslash \{ -1,0 \}, \\ \vartheta , & \vartheta =\varpi. \end{cases}\displaystyle \quad \vartheta ,\varpi >0. $$ -
6.
The identric mean
$$ I:=I(\vartheta ,\varpi )=\frac{1}{e} \biggl( \frac{\varpi ^{\varpi }}{\vartheta ^{\vartheta }} \biggr) ^{\frac{1}{\varpi -\vartheta }},\quad \vartheta ,\varpi >0. $$
The following simple relationships are known in the literature:
Proposition 1
Let \(\vartheta ,\varpi \in [ -1,1 ] \) with \(\vartheta <\varpi \). Then,
Proof
The assertion follows from (3.1) for the function
□
Proposition 2
Let \(\vartheta ,\varpi \in [ -1,1 ] \) with \(\vartheta <\varpi \). Then,
Proof
The proof is easily obtained from (3.1) for the function \(\Lambda {(\sigma )}=\frac{2}{3}{\sigma }^{\frac{3}{2}}\) because the function
is GSPOLC. □
Proposition 3
Let \(\vartheta ,\varpi \in [ -1,1 ] \) with \(\vartheta <\varpi \) and \(q>1\). Then,
Proof
The proof is easily seen from (3.1) for the function \(\Lambda {(\sigma )}=\frac{q}{2+q}{\sigma }^{\frac{2+q}{q}}\) because the function
is GSPOLC. □
Proposition 4
Let \(\vartheta ,\varpi \in [ -1,1 ] \) with \(\vartheta <\varpi \) and \(q\geq 1\). Then,
Proof
The proof is obvious from (3.1) for the function \(\Lambda {(\sigma )}=\frac{q}{2+q}{\sigma }^{\frac{2+q}{q}}\) because the function
is GSPOLC. □
Proposition 5
Let \(\vartheta ,\varpi \in [ -1,1 ] \) with \(\vartheta <\varpi \) and \(q\geq 1\). Then,
Proof
The proof is easily seen from (3.1) for the function \(\Lambda {(\sigma )}=\frac{q}{2+q}{\sigma }^{\frac{2+q}{q}}\) because the function
is GSPOLC. □
Proposition 6
Let \(\vartheta ,\varpi \in [ -1,1 ] \) with \(\vartheta <\varpi \) and \(q\geq 1\). Then,
Proof
The proof is easily obtained from (3.1) for the function \(\Lambda {(\sigma )}=\frac{q}{2+q}{\sigma }^{\frac{2+q}{q}}\) because the function
is GSPOLC. □
6 Conclusion
This paper introduces a novel class of functions called generalized strongly n-polynomial convex functions. The relationships between these functions and other types of convex functions are investigated. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of the Hermite–Hadamard type are derived for this specific function class using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with existing findings in the literature, highlighting the superiority of the newly acquired results. Furthermore, some applications of these results to special means are explored.
Data availability
No datasets were generated or analysed during the current study.
References
Abdeljawad, T., Mohammed, P.O., Kashuri, A.: New modified comformable fractional integral inequalities of Hermite-Hadamard type with applications. J. Funct. Spaces 2020, Article ID 4352357 (2020)
Akdemir, A.O., Aslan, S., Dokuyucu, M.A., Çelik, E.: Exponentially convex functions on the coordinates and novel estimations via Riemann-Liouville fractional operator. J. Funct. Spaces 2023, Article ID 4310880 (2023)
Angulo, H., Gimenez, J., Moros, A.M., Nikodem, K.: On strongly h-convex functions. Ann. Funct. Anal. 2(2), 85–91 (2011)
Ataman, C., Kadakal, M., İşcan, İ.: Strongly n-polynomial convexity and related inequalities. Creative Math. Inform. 31(2), 155–172 (2022)
Bombardelli, M., Varosanec, S.: Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities. Comput. Math. Appl. 58, 1869–1877 (2009)
Butt, S.I., Pecaric, J.: Generalized Hermite-Hadamard’s inequality. Proc. A. Razmadze Math. Inst. 163, 9–27 (2013)
Butt, S.I., Tariq, M., Aslam, A., Ahmad, H., Nofal, T.A.: Hermite-Hadamard type inequalities via generalized harmonic exponential convexity and applications. J. Funct. Spaces 2021, Article ID 5533491 (2021)
Chasreechai, S., Ali, M.A., Naowarat, S., Sitthiwirattham, T., Nonlaopon, K.: On some Simpson’s and Newton’s type of inequalities in multiplicative calculus with applications. AIMS Math. 8(2), 3885–3896 (2023)
Dragomir, S.S., Agarwal, R.P.: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 11, 91–95 (1998)
Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite-Hadamard Inequalities and Its Applications. RGMIA Monograph (2002)
Dragomir, S.S., Pecaric, J., Persson, L.E.: Some inequalities of Hadamard type. Soochow J. Math. 21(3), 335–341 (2001)
Hadamard, J.: Étude sur les propriétés des fonctions entières en particulier d’une fonction considérée par Riemann. J. Math. Pures Appl. 58, 171–215 (1893)
İşcan, İ.: New refinements for integral and sum forms of Hölder inequality. J. Inequal. Appl. 2019(1), 304 (2019)
İşcan, İ., Kadakal, H., Kadakal, M.: Some new integral inequalities for n-times differentiable quasi-convex functions. Sigma J. Eng. Nat. Sci. 35(3), 363–368 (2017)
İşcan, İ., Kunt, M.: Hermite-Hadamard-Fejer type inequalities for quasi-geometrically convex functions via fractional integrals. J. Math. 2016, Article ID 6523041 (2016)
Kadakal, H.: New inequalities for strongly r-convex functions. J. Funct. Spaces 2019, Article ID 1219237 (2019)
Kadakal, H., Kadakal, M., İşcan, İ.: New type integral inequalities for three times differentiable preinvex and prequasiinvex functions. Open J. Math. Anal. 2(1), 33–46 (2018)
Kadakal, M., İşcan, İ., Kadakal, H.: Construction of a new generalization for n-polynomial convexity with their certain inequalities. Hacet. J. Math. Stat. (2023). https://doi.org/10.15672/hujms.xx
Kadakal, M., İşcan, İ., Kadakal, H., Bekar, K.: On improvements of some integral inequalities. Honam Math. J. 43(3), 441–452 (2021)
Latif, M.A., Kunt, M., Dragomir, S.S., İşcan, İ.: Post-quantum trapezoid type inequalities. AIMS Math. 5(4), 4011–4026 (2020)
Maden, S., Kadakal, H., Kadakal, M., İşcan, İ.: Some new integral inequalities for n-times differentiable convex and concave functions. J. Nonlinear Sci. Appl. 10(12), 6141–6148 (2017)
Merentes, N., Nikodem, K.: Remarks on strongly convex functions. Aequ. Math. 80(1–2), 193–199 (2010)
Özcan, S.: On refinements of some integral inequalities for differentiable prequasiinvex functions. Filomat 33(14), 4377–4385 (2019)
Özcan, S.: Some integral inequalities of Hermite-Hadamard type for multiplicatively s-preinvex functions. Int. J. Math. Model. Comput. 9(4), 253–266 (2019)
Özcan, S.: Hermite-Hadamard type inequalities for m-convex and \((\alpha , m)\)-convex functions. J. Inequal. Appl. 2020(1), 175 (2020)
Özcan, S.: Hermite-Hadamard type inequalities for multiplicatively s-convex functions. Cumhuriyet Sci. J. 41(1), 245–259 (2020)
Özcan, S.: Hermite-Hadamard type inequalities for exponential type multiplicatively convex functions. Filomat 37(28), 9777–9789 (2023)
Pearce, C.E.M., Pecaric, J.: Inequalities for differentiable mappings with application to special means and quadrature formulae. Appl. Math. Lett. 13, 51–55 (2000)
Polyak, P.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restictions. Sov. Math. Dokl. 7, 72–75 (1966)
Toplu, T., Kadakal, M., İşcan, İ.: On n-polynomial convexity and some related inequalities. AIMS Math. 5(2), 1304 (2020)
Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326, 303–311 (2007)
Vivas-Cortez, M., Kara, H., Budak, H., Ali, M.A., Chasreechai, S.: Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions. Open Math. 20(1), 1887–1903 (2022)
Vivas-Cortez, M., Kashuri, A., Liko, R., Hernandez Hernandez, J.E.: Some inequalities using generalized convex functions in quantum analysis. Symmetry 11, 1402–1426 (2019)
Xie, J., Ali, M.A., Budak, H., Fečkan, M., Sitthiwirattham, T.: Fractional Hermite-Hadamard inequality, Simpson’s and Ostrowski’s type inequalities for convex functions with respect to a pair of functions. Rocky Mt. J. Math. 53(2), 611–628 (2023)
Zabandan, G.: A new refinement of the Hermite-Hadamard inequality for convex functions. J. Inequal. Pure Appl. Math. 10(2), Article ID 45 (2009)
Acknowledgements
Authors are thankful to editor and anonymous referees for their valuable comments and suggestions.
Funding
There is no funding for this research article.
Author information
Authors and Affiliations
Contributions
SÖ and MK gave the idea and initiated the writing of this paper. İİ and HK followed up this with some complementary ideas. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Özcan, S., Kadakal, M., İşcan, İ. et al. Generalized strongly n-polynomial convex functions and related inequalities. Bound Value Probl 2024, 32 (2024). https://doi.org/10.1186/s13661-024-01838-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-024-01838-2