Opial obtained the following integral inequality in 1960 [25].
Theorem 1.1
Let
\(g\in C^{1}[0, h]\)
be such that
\(g(0)=g(h)=0\)
and
\(g(t)>0\)
for
\(t \in (0, h)\). Then
$$ \int _{0}^{h} \bigl\vert g(t)g^{\prime }(t) \bigr\vert \,dt \le \frac{h}{4} \int _{0}^{h}\bigl(g'(t) \bigr)^{2}\,dt. $$
Here
\(\frac{h}{4}\)
is a best possible constant.
This inequality has been studied by several mathematicians, and several related inequalities have been investigated. For example its generalizations were published by Beesack in 1962 and 1971 [3, 5], Hua in 1965 [17], Redheffer in 1966 [30], Calvert in 1967 [8], Godunova and Levin in 1967 [13], Maroni in 1967 [22], Boyd and Wong in 1967 [7], Beesack and Das in 1968 [4], Boyd in 1969 [6], Rozanova in 1972 [31], Vrǎnceanu in 1973 [35], Shum in 1974 and 1975 [32, 33], Hou in 1979 [16], G. Milovanovic and I. Milovanovic in 1980 [23], Lee in 1980 [20], He and Wang in 1981 [14], Yang in 1966 and 1983 [37, 38], Hong, Yang, and Du in 1982 [15], Lin and Yang in 1985 [21], Qi in 1985 [29], Fagbohun and Imoru in 1985 and 1986 [9, 10], Pachpatte in 1986 and 1993 [26, 27], Mitrinović and Pečarić in 1988 [24], Hwang and Yang in 1990 [18], and Sinnamon in 1991 [34].
The aim of this paper is to establish some new Opial-type inequalities for convex functions. Therefore, in the subsequent convex function, its properties, characterization, and Opial-type inequalities for convex functions have been summarized as motivation behind the recent work.
Definition 1
Let I be an interval in \(\mathbb{R}\). Then \(f : I \to \mathbb{R}\) is said to be convex if, for all \(x, y \in I\) and all \(\alpha \in [0, 1]\),
$$ f\bigl(\alpha x +(1 - \alpha )y\bigr) \le \alpha f(x) + (1-\alpha )f(y) $$
(1)
holds.
A characterization of convex function is stated in the following lemma.
Lemma 1.2
([36])
Let
f
be a differentiable function on
\((a, b)\). Then
f
is convex if and only if
\(f^{\prime }\)
is an increasing function.
The convexity of composition of two functions can be obtained under the conditions stated in the following lemma.
Lemma 1.3
([36])
Let
\(f : I \to \mathbb{R}\)
and
\(g : J \to \mathbb{R}\), where
\(\operatorname{range}(f) \subseteq J\). If
f
and
g
are convex and
g
is increasing, then the composite function
\(g \circ f\)
is convex on
I.
In [24] generalized Opial-type inequalities for convex functions have been proved by Mitrinović and Pečarić. Let \(U_{1}(v, k)\) denote the class of functions \(u : [a, b]\to \mathbb{R}\) having representation
$$ u(x)= \int _{a}^{x}k(x, t)v(t)\,dt, $$
where v is a continuous function and k is an arbitrary nonnegative kernel such that \(k(x, t)=0\) for \(t>x\), and \(v(x) > 0\) implies \(u(x) > 0\) for every \(x \in [a, b]\).
Let \(U_{2}(v, k)\) denote the class of all the functions \(u : [a, b] \to \mathbb{R}\) having representation
$$ u(x)= \int _{x}^{b}k(x, t)v(t)\,dt, $$
where v is a continuous function and k is an arbitrary nonnegative kernel such that \(k(x, t)=0\) for \(t< x\), and \(v(x) > 0\) implies \(u(x) > 0\) for every \(x \in [a, b]\).
Theorem 1.4
([28])
Let
\(\phi : [0, \infty ) \to \mathbb{R}\)
be a differentiable function such that, for
\(q > 1\), the function
\(\phi (x^{1/q})\)
is convex and
\(\phi (0) = 0\). Let
\(u \in U_{1}(v, k)\)
where
\(( \int _{a} ^{x}(k(x, t))^{p}\,dt )^{1/p} \leq K\)
and
\(\frac{1}{p} + \frac{1}{q} = 1\). Then
$$ \int _{a}^{b} \bigl\vert u(x) \bigr\vert ^{1-q}\phi ^{\prime }\bigl( \bigl\vert u(x) \bigr\vert \bigr) \bigl\vert v(x) \bigr\vert ^{q}\,dx \le \frac{q}{K ^{q}}\phi \biggl( K \biggl( \int _{a}^{b} \bigl\vert v(x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \biggr). $$
If the function
\(\phi (x^{1/q})\)
is concave, then the reverse inequality holds.
A similar result was obtained for class \(U_{2}(v, k)\).
Theorem 1.5
([28])
Let
\(\phi : [0, \infty ) \to \mathbb{R}\)
be a differentiable function such that, for
\(q > 1\), the function
\(\phi (x^{1/q})\)
is convex and
\(\phi (0) = 0\). Let
\(u \in U_{2}(v, k)\)
where
\(( \int _{x} ^{b}(k(x, t))^{p}\,dt )^{1/p} \leq K\)
and
\(\frac{1}{p} + \frac{1}{q} = 1\). Then
$$ \int _{a}^{b} \bigl\vert u(x) \bigr\vert ^{1-q}\phi ^{\prime }\bigl( \bigl\vert u(x) \bigr\vert \bigr) \bigl\vert v(x) \bigr\vert ^{q}\,dx \le \frac{q}{K ^{q}}\phi \biggl( K \biggl( \int _{a}^{b} \bigl\vert v(x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \biggr). $$
If the function
\(\phi (x^{1/q})\)
is concave, then the reverse inequality holds.
In [1] Andrić et al. further extended these results stating the following.
Theorem 1.6
([1])
Let
\(\phi : [0, \infty ) \to \mathbb{R}\)
be a differentiable function such that, for
\(q > 1\), the function
\(\phi (x^{1/q})\)
is convex and
\(\phi (0) = 0\). Let
\(u \in U_{1}(v, k)\)
where
\(( \int _{a} ^{x}(k(x, t))^{p}\,dt )^{1/p} \leq K\)
and
\(\frac{1}{p} + \frac{1}{q} = 1\). Then
$$\begin{aligned} \int _{a}^{b} \bigl\vert u(x) \bigr\vert ^{1-q}\phi ^{\prime }\bigl( \bigl\vert u(x) \bigr\vert \bigr) \bigl\vert v(x) \bigr\vert ^{q}\,dx &\le \frac{q}{K ^{q}}\phi \biggl( K \biggl( \int _{a}^{b} \bigl\vert v(x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \biggr) \\ & \le \frac{q}{K^{q}(b-a)} \int _{a}^{b}\phi \bigl((b-a)^{1/q}K \bigl\vert v(x) \bigr\vert \bigr)\,dx. \end{aligned}$$
If the function
\(\phi (x^{1/q})\)
is concave, then the reverse inequality holds.
A similar result for class \(U_{2}(v, k)\) is stated in the following theorem.
Theorem 1.7
([1])
Let
\(\phi : [0, \infty ) \to \mathbb{R}\)
be a differentiable function such that, for
\(q > 1\), the function
\(\phi (x^{1/q})\)
is convex and
\(\phi (0) = 0\). Let
\(u \in U_{2}(v, k)\)
where
\(( \int _{x} ^{b}(k(x, t))^{p}\,dt )^{1/p} \leq K\)
and
\(\frac{1}{p} + \frac{1}{q} = 1\). Then
$$\begin{aligned} \int _{a}^{b} \bigl\vert u(x) \bigr\vert ^{1-q}\phi ^{\prime }\bigl( \bigl\vert u(x) \bigr\vert \bigr) \bigl\vert v(x) \bigr\vert ^{q}\,dx &\le \frac{q}{K ^{q}}\phi \biggl( K \biggl( \int _{a}^{b} \bigl\vert v(x) \bigr\vert ^{q}\,dx \biggr)^{1/q} \biggr) \\ & \le \frac{q}{K^{q}(b-a)} \int _{a}^{b}\phi \bigl((b-a)^{1/q}K \bigl\vert v(x) \bigr\vert \bigr)\,dx. \end{aligned}$$
If the function
\(\phi (x^{1/q})\)
is concave, then the reverse inequality holds.
In [11, 12] Farid and Pečarić studied these inequalities in a fractional point of view. They considered several fractional integral operators via particular kernels to obtain Riemann–Liouville, Caputo, and Canavati fractional Opial-type inequalities. Extensions of these Opial-type fractional inequalities have been proved in [1] by Andrić et al., Basci and Dumitru in [2] considered some new aspects of these inequalities.
Next we present the Riemann–Liouville fractional integral, Caputo, and Canavati fractional derivatives [19].
Definition 2
Let \(f \in L_{1}[a, b]\). Then the left-sided and right-sided Riemann–Liouville fractional integrals of order \(\alpha > 0\) with \(a \ge 0\) are defined as follows:
$$ I^{\alpha }_{a+}f(x) = \frac{1}{\varGamma (\alpha )} \int _{a}^{x}(x-t)^{ \alpha -1}f(t)\,dt, \quad x > a, $$
and
$$ I^{\alpha }_{b-}f(x) = \frac{1}{\varGamma (\alpha )} \int _{x}^{b}(t-x)^{ \alpha -1}f(t)\,dt, \quad x < b, $$
where \(\varGamma (\cdot)\) is the gamma function.
Definition 3
Let \(\alpha >0\) and \(\alpha \notin \{1, 2, 3, \ldots \}\), \(n = [\alpha ]+1\), \(f \in AC^{n}[a,b]\). Then the left-sided and right-sided Caputo fractional derivatives of order α are defined as follows:
$$\begin{aligned} & \bigl( {^{C}}D^{\alpha }_{a+}f \bigr) (x)= \frac{1}{\varGamma (n- \alpha )} \int ^{x}_{a}\frac{f^{(n)} {(t)}}{(x-t)^{{\alpha }-n+1}}\,dt ,\quad x>a, \end{aligned}$$
(2)
and
$$\begin{aligned} & \bigl( {^{C}}D^{\alpha }_{b-}f \bigr) (x)= \frac{(-1)^{n}}{\varGamma (n- \alpha )} \int ^{b}_{x}\frac{f^{(n)} {(t)}}{(t-x)^{{\alpha }-n+1}}\,dt ,\quad x< b. \end{aligned}$$
(3)
In [1] composition identities for the Caputo fractional derivatives are given, they are stated in the following lemmas.
Lemma 1.8
Let
\(\beta > \alpha \ge 0\), \(m = [\beta ]+1\), and
\(n = [\alpha ]+1\)
for
\(\alpha , \beta \notin \mathbb{N}_{0}\); \(n = [\alpha ]\)
and
\(m = [\beta ]\)
for
\(\alpha , \beta \in \mathbb{N}_{0}\). Let
\(f \in AC^{m} [a, b]\)
be such that
\(f^{(i)}(a) = 0\)
for
\(i = n, n+1, \ldots , m-1\). Let
\({}^{C}D^{\beta }_{a+}f, ^{C}D^{\alpha }_{a+}f \in {L_{1}[a, b]}\). Then
$$ ^{C}D^{\alpha }_{a+}f(x) = \frac{1}{\varGamma (\beta - \alpha )} \int _{a} ^{x}(x-t)^{\beta -\alpha -1} {} ^{C}D^{\beta }_{a+}f(t)\,dt, \quad x \in [a, b]. $$
Lemma 1.9
Let
\(\beta > \alpha \ge 0\), \(m = [\beta ]+1\), and
\(n = [\alpha ]+1\)
for
\(\alpha , \beta \notin \mathbb{N}_{0}\); \(n = [\alpha ]\)
and
\(m = [\beta ]\)
for
\(\alpha , \beta \in \mathbb{N}_{0}\). Let
\(f \in AC^{m} [a, b]\)
be such that
\(f^{(i)}(b) = 0\)
for
\(i = n, n+1, \ldots , m-1\). Let
\({}^{C}D^{\alpha }_{b-}f, ^{C}D^{\beta }_{b-}f \in {L_{1}[a, b]}\). Then
$$ ^{C}D^{\alpha }_{b-}f(x) = \frac{1}{\varGamma (\beta - \alpha )} \int _{x} ^{b}(t-x)^{\beta -\alpha -1} {} ^{C}D^{\beta }_{b-}f(t)\,dt, \quad x \in [a, b]. $$
Next consider a subspace \(C^{\alpha }_{a+}[a, b]\) defined by
$$ C^{\alpha }_{a+}[a, b] = \bigl\{ f \in C^{n-1}[a, b] : I^{n-\alpha }_{a+}f ^{(n-1)} \in C^{1}[a, b]\bigr\} . $$
Definition 4
Let \(f \in C^{\alpha }_{a+}[a, b]\). Then the left-sided Canavati fractional derivative is defined by
$$ ^{\tilde{C}}D^{\alpha }_{a+}f(x) = \frac{1}{\varGamma (n-\alpha )} \frac{d}{dx} \int _{a}^{x}(x-t)^{n-\alpha -1}f^{(n-1)}(t) \,dt = \frac{d}{dx}I^{n-\alpha }_{a+}f^{(n-1)}(x). $$
Composition identity for the left-sided Canavati fractional derivative is given in the following lemma.
Lemma 1.10
([1])
Let
\(\beta > \alpha > 0\), \(m = [\beta ]+1\), \(n = [\alpha ]+1\). Let
\(f \in C^{\beta }_{a+}[a, b]\)
be such that
\(f^{i}(a) = 0\)
for
\(i = n-1, n,\ldots, m-2\). Then
\(f \in C^{\alpha }_{a+}[a, b]\)
and
$$ {}^{\tilde{C}}D^{\alpha } _{a+}f(x)=\frac{1}{\varGamma (\beta -\alpha )} \int _{a}^{x}(x-t)^{\beta -\alpha -1} {}^{\tilde{C}}D^{\beta }_{a+}f(t)\,dt,\quad x \in [a, b]. $$
Definition 5
Let \(f\in L_{1}[a, b]\). Then the left-sided Riemann–Liouville fractional derivative of order α is defined by
$$ D^{\alpha }_{a+}f(x) = \frac{1}{\varGamma (n-\alpha )}\frac{d^{n}}{dx ^{n}} \int _{a}^{x}(x-t)^{n-\alpha -1}f(t)\,dt = \frac{d^{n}}{dx^{n}}I ^{n-\alpha }_{a+}f(x). $$
(4)
The following lemma summarizes conditions in the composition identity for the left-sided Riemann–Liouville fractional derivative.
Lemma 1.11
([1])
Let
\(\beta > \alpha \ge 0\), \(m = [\beta ]+1\), \(n = [ \alpha ]+1\). The composition identity
$$ D^{\alpha }_{a+}f(x) = \frac{1}{\varGamma (\beta -\alpha )} \int _{a}^{x}(x-t)^{ \beta -\alpha -1}D^{\beta }_{a+}f(t) \,dt, \quad x \in [a, b] $$
(5)
is valid if one of the following conditions holds:
-
(i)
\(f\in I^{\beta }_{a+}(L_{1}[a, b]) = \{f : f = I^{\beta }_{a+}\varphi ,\varphi \in L_{1}[a, b] \}\).
-
(ii)
\(I^{m-\beta }_{a+}f \in AC^{m}[a, b]\)
and
\(D^{\beta -k}_{a+}f(a) = 0\)
for
\(k=1,\ldots,m\).
-
(iii)
\(D^{\beta -1}_{a+}f \in AC[a, b]\), \(D^{\beta -k}_{a+}f \in C[a, b]\)
and
\(D^{\beta -k}_{a+}f(a) = 0\)
for
\(k = 1,\ldots,m\).
-
(iv)
\(f \in AC^{m}[a, b]\), \(D^{\beta }_{a+}f, D^{\alpha }_{a+}f \in L_{1}[a, b]\), \(\beta -\alpha \notin \mathbb{N}, D^{\beta -k}_{a+}f(a) = 0\)
for
\(k =1,\ldots,m\)
and
\(D^{\alpha -k}_{a+}f(a) = 0\)
for
\(k = 1,\ldots,n\).
-
(v)
\(f \in AC^{m}[a, b]\), \(D^{\beta }_{a+}f, D^{\alpha }_{a+}f \in L_{1}[a, b]\), \(\beta -\alpha = l \in \mathbb{N}\), \(D^{\beta -k}_{a+}f(a) = 0\)
for
\(k =1,\ldots,l\).
-
(vi)
\(f \in AC^{m}[a, b]\), \(D^{\beta }_{a+}f, D^{\alpha }_{a+}f \in L_{1}[a, b]\)
and
\(f^{k}(a) = 0\)
for
\(k =0,\ldots,m-2\).
-
(vii)
\(f \in AC^{m}[a, b]\), \(D^{\beta }_{a+}f, D^{\alpha }_{a+}f \in L_{1}[a, b]\), \(\beta \notin \mathbb{N}\)
and
\(D^{\beta -1}_{a+}f\)
is bounded in a neighborhood of
a.
In the following paragraph, the spaces of functions which have been used in definitions and results are summarized.
The space of all continuous functions whose nth time continuous derivative exists on \([a, b]\) is denoted by \(C^{n}[a, b]\), the space of all absolutely continuous functions on \([a, b]\) is denoted by \(AC[a, b]\). While \(AC^{n}[a, b]\) denotes the space of all the functions \(f \in C^{n-1}[a, b]\) with \(f^{(n-1)} \in AC[a, b]\) and \(L_{p}[a, b]\), \(1 \leq p < \infty \), denotes the space of all Lebesgue measurable functions f for which \(\vert f \vert ^{p}\) is Lebesgue integrable on \([a, b]\).
The paper is organized as follows.
In Sect. 2, new Opial-type inequalities for convex functions are established by applying an arbitrary kernel. Moreover, these inequalities are studied for a power function. Furthermore, in Sect. 3 results of Sect. 2 are analyzed for particular kernels, and fractional integral inequalities of Opial-type are produced by using the definitions and composition identities of Riemann–Liouville fractional integral, Caputo fractional derivative, Canavati fractional derivative. The results for fractional inequalities are obtained by using different forms of the weighted functions and kernels.