Effects of teaching approaches on science subject choice toward STEM career orientation of Vietnamese students

1.   Introduction

Let $I\subseteq \mathbb{R}$ be an interval. Then a real-valued function $h: I\rightarrow \mathbb{R}$ is said to be convex (concave) on the interval $I$ if the inequality

holds for all $\kappa_{1}, \kappa_{2}\in I$ and $t\in [0, 1]$.

It is well known that convexity (concavity) has wide applications in pure and applied mathematics ^{[1,2,3,4,5,6,7,8,9,10,11,12]}. The well known Hermite-Hadamard inequality ^{[13,14,15,16,17,18,19,20]} for the convex (concave) function $h: I\rightarrow \mathbb{R}$ can be stated as follows:

for all $\kappa_{1}, \kappa_{2}\in I$ with $\kappa_{1}\neq \kappa_{2}$.

Recently, many generalizations, invariants and extensions have been made for the convexity, for example, harmonic-convexity ^[21,22], exponential-convexity ^[23,24], $s$-convexity ^[25,26], Schur-convexity ^[27,28,29], strong convexity ^{[30,31,32,33]}, $H_{p, q}$-convexity ^{[34,35,36,37,38]}, generalized convexity ^[39], $GG$- and $GA$-convexities ^[40], preinvexity ^[41] and quasi-convexity ^[42]. In particular, many remarkable inequalities can be found in the literature ^{[43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58]} via the convexity theory.

Niculescu ^[59,60] defined the $GG$- and $GA$-convex functions as follows.

Definition 1.1. (See ^[59]) A real-valued function $h: I\rightarrow [0, \infty)$ is said to be $GG$-convex on the interval $I$ if the inequality

holds for all $\kappa_{1}, \kappa_{2}\in I$ and $t\in[0, 1]$.

Definition 1.2. (See ^[60]) A real-valued function $h : I\rightarrow [0, \infty)$ is said to be $GA$-convex if the inequality

holds for all $\kappa_{1}, \kappa_{2}\in I$ and $t\in[0, 1]$.

Ardıç et al. ^[61] established several novel inequalities (Theorem 1.1) involving the $GG$- and $GA$-convex functions via an identity (Lemma 1.1) for differentiable functions.

Lemma 1.1. (See ^[61]) Let $\kappa_{1}, \kappa_{2}\in (0, \infty)$ with $\kappa_{1} < \kappa_{2}$ and $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be a differentiable function such that $h^{\prime}\in L([\kappa_{1}, \kappa_{2}])$. Then the identity

holds for all $\eta\in[\kappa_{1}, \kappa_{2}]$.

Theorem 1.1. (See ^[61]) Let $\kappa_{1}, \kappa_{2}\in (0, \infty)$ with $\kappa_{1} < \kappa_{2}$ and $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be a differentiable function such that $h^{\prime}\in L([\kappa_{1}, \kappa_{2}])$. Then the following statements are true:

$(1)$ If $|h^{\prime}(x)|$ is $GG$-convex on $[\kappa_{1}, \kappa_{2}]$, then the inequality

holds for all $\eta\in[\kappa_{1}, \kappa_{2}]$, where $L(\kappa_{1}, \kappa_{2}) = (\kappa_{2}-\kappa_{1})/(\log\kappa_{2}-\log\kappa_{1})$ is the logarithmic mean of $\kappa_{1}$ and $\kappa_{2}$.

$(2)$ If $\vartheta, \gamma > 1$ with $1/\vartheta+1/\gamma = 1$ and $|h^{\prime}(x)|^{\gamma}$ is $GG$-convex on $[\kappa_{1}, \kappa_{2}]$, then the inequalities

and

hold for all $\eta\in[\kappa_{1}, \kappa_{2}]$.

$(3)$ If $|h^{\prime}(x)|$ is $GA$-convex on $[\kappa_{1}, \kappa_{2}]$, then we have

for all $\eta\in[\kappa_{1}, \kappa_{2}]$.

$(4)$ If $\vartheta, \gamma > 1$ with $1/\vartheta+1/\gamma = 1$ and $|h^{\prime}(x)|^{\gamma}$ is $GA$-convex on $[\kappa_{1}, \kappa_{2}]$, then one has

where

and

The conformable fractional derivative $D_{\alpha}(h)(t)$ ^[62] of order $0 < \alpha\leq1$ at $t > 0$ for a function $h: [0, \infty)\rightarrow \mathbb{R}$ is defined by

$h$ is said to be $\alpha$-fractional differentiable if the conformable fractional derivative $D_{\alpha}(h)(t)$ exists. The conformable fractional derivative at $0$ is defined by $h^{\alpha}(0) = \lim_{t \rightarrow 0^{+}}h^{\alpha}(t)$. If $h_{1}$ and $h_{2}$ are $\alpha$-differentiable at $t > 0$, and $\kappa_{1}, \kappa_{2}, \lambda, c\in\mathbb{R}$ are constants, then the conformable fractional derivative satisfies the following formulas

and

if $h_{1}$ differentiable at $h_{2}(t)$. Moreover,

if $h_{1}$ is differentiable.

Let $\alpha\in(0, 1]$ and $0\leq \kappa_{1} < \kappa_{2}$. Then the function $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ is said to be $\alpha$-fractional integrable on $[\kappa_{1}, \kappa_{2}]$ if the integral

exists and is finite. All $\alpha$-fractional integrable functions on $[\kappa_{1}, \kappa_{2}]$ is denoted by $L_{\alpha}([\kappa_{1}, \kappa_{2}])$. Note that

for all $\alpha\in (0, 1]$, where the integral is the usual Riemann improper integral.

Recently, the conformable integrals and derivatives have attracted the attention of many researchers. Anderson ^[63] established the conformable integral version of the Hermite-Hadamard inequality as follows:

if $\alpha\in(0, 1]$ and $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ is an $\alpha$-fractional differentiable function such that $D_{\alpha}(h)$ is increasing. Moreover, if function $h$ is decreasing on $[\kappa_{1}, \kappa_{2}]$, then

The main purpose of the article is to establish the conformable fractional integral versions of the Hermite-Hadamard type inequality for $GG$- and $GA$-convex functions.

2.   Main results

In order to establish our main results, we need a lemma which we present in this section.

Lemma 2.1. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\alpha \in (0, 1]$ and $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$. Then the identity

holds for all $\eta\in[\kappa_{1}, \kappa_{2}]$.

Proof. Integration by parts, we get

Let $x = \kappa_{2}^{t}\eta^{1-t}$. Then $I_{1}$ can be rewritten as

Similarly, we have

Multiplying $I_{1}$ by $(\log\kappa_{2}-\log\eta)$ and $I_{2}$ by $(\log \eta-\log\kappa_{1})$, then add them we get the desired identity.

Remark 2.1. Let $\alpha = 1$. Then identity (2.1) reduces to (1.1).

Theorem 2.1. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\alpha \in (0, 1]$, $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|$ be a $GG$-convex function on $[\kappa_{1}, \kappa_{2}]$. Then the inequality

holds for all $\eta\in [\kappa_{1}, \kappa_{2}]$.

Proof. It follows from the $GG$-convexity of the function $|h^{\prime}(x)|$ on the interval $[\kappa_{1}, \kappa_{2}]$ and Lemma 2.1 that

Remark 2.2. Let $\alpha = 1$. Then inequality (2.2) reduces to (1.2).

Theorem 2.2. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\vartheta, \gamma > 1$ with $1/\vartheta+1/\gamma = 1$, $\alpha \in (0, 1]$, $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GG$-convex function on $[\kappa_{1}, \kappa_{2}]$. Then the inequality

holds for all $\eta\in [\kappa_{1}, \kappa_{2}]$.

Proof. From Lemma 2.1, the property of the modulus, $GG$-convexity of $|h^{\prime}|^\gamma$ and Hölder inequality we clearly see that

Remark 2.3. Let $\alpha = 1$. Then inequality (2.3) reduces to (1.3).

Theorem 2.3. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\gamma > 1$, $\alpha\in (0, 1]$, $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GG$-convex function on $[\kappa_{1}, \kappa_{2}]$. Then the inequality

holds for all $\eta\in [\kappa_{1}, \kappa_{2}]$.

Proof. It follows from Lemma 2.1 that

Let $\vartheta > 1$ such that $\vartheta^{-1}+\gamma^{-1} = 1$. Then making use of the Hölder integral inequality and the $GG$-convexity of $|h^{\prime}|^{\gamma}$, we get

Remark 2.4. Let $\alpha = 1$. Then inequality (2.4) reduces to (1.4).

Theorem 2.4. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\gamma > 1$, $\alpha \in (0, 1]$, $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GG$-convex function on $[\kappa_{1}, \kappa_{2}]$. Then the inequality

holds whenever $\eta\in [\kappa_{1}, \kappa_{2}]$.

Proof. From the $GG$-convexity of $|h^{\prime}|^{\gamma}$, power mean inequality, the property of the modulus and Lemma 2.1 we clearly see that

Remark 2.5. Let $\alpha = 1$. Then inequality (2.5) reduces to (1.5).

Theorem 2.5. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\alpha \in (0, 1]$, $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|$ be a $GA$-convex function on $[\kappa_{1}, \kappa_{2}]$. Then the inequality

holds for each $\eta\in [\kappa_{1}, \kappa_{2}]$.

Proof. It follows from the $GA$-convexity of $|h^{\prime}(x)|$ and Lemma 2.1 that

Remark 2.6. Let $\alpha = 1$. Then inequality (2.6) becomes (1.6).

Theorem 2.6. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\alpha \in (0, 1]$, $\gamma > 1$, $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GA$-convex function on $[\kappa_{1}, \kappa_{2}]$. Then the inequality

holds for any $\eta\in [\kappa_{1}, \kappa_{2}]$.

Proof. From the $GA$-convexity of $|h^{\prime}|^{\gamma}$, power mean inequality, the property of the modulus and Lemma 2.1, one has

Remark 2.7. Let $\alpha = 1$. Then inequality (2.7) reduces to (1.7).

Theorem 2.7. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\vartheta, \gamma > 1$ with $1/\vartheta+1/\gamma = 1$, $\alpha \in (0, 1]$, $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D_{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GA$-convex function on $[\kappa_{1}, \kappa_{2}]$. Then the inequality

holds for any $\eta\in [\kappa_{1}, \kappa_{2}]$, where

Proof. It follows from Lemma 2.1, the $GA$-convexity of $|h^{\prime}|^\gamma$, power mean inequality, Hölder integral inequality and the property of the modulus that

Remark 2.8. Let $\alpha = 1$. Then inequality (2.8) becomes (1.8).

Theorem 2.8. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\gamma > 1$, $\alpha \in (0, 1]$, $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GA$-convex function on $[\kappa_{1}, \kappa_{2}]$. Then the inequality

holds for any $\eta\in [\kappa_{1}, \kappa_{2}]$.

Proof. From Lemma 2.1, the $GG$-convexity of $|h^{\prime}|^{\gamma}$, Hölder inequality and the property of the modulus, we have

Remark 2.9. Let $\alpha = 1$. Then inequality (2.9) leads to (1.9).

Theorem 2.9. Let $\kappa_{1}, \kappa_{2}\in(0, \infty)$ with $\kappa_{1} < \kappa_{2}$, $\gamma > 1$, $\alpha\in (0, 1]$, $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\alpha$-fractional differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\alpha}(h)\in L_{\alpha}([\kappa_{1}, \kappa_{2}])$ and $|h^{\prime}(x)|^{\gamma}$ be a $GA$-convex function on $[\kappa_{1}, \kappa_{2}]$. Then the inequality

holds for any $\eta\in [\kappa_{1}, \kappa_{2}]$, where

Proof. It follows from Lemma 2.1, the $GA$-convexity of $|h^{\prime}|^\gamma$, power mean inequality and property of the modulus that

Remark 2.10. Let $\alpha = 1$. Then inequality (2.10) reduces to (1.10).

3.   Conclusion

We have generalized the Hermite-Hadamard type inequalities for $GG$- and $GA$-convex functions established by Ardıç, Akdemir and Yıdız in ^[61] to the conformable fractional integrals. Our ideas and approach may lead to a lot of follow-up research.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11971142, 11701176, 11626101, 11601485).

Conflict of interest

The authors declare no conflict of interest.