To prove our main result, we need the following definition and lemma.
Definition 1
(Polyak
1996) Let \(f:I\subseteq R\rightarrow R\) is said to be strongly s-convex with modulus \(c>0\) and for some fixed \(s\in \left( {0,1} \right]\), if
$$\begin{aligned} f\left( {\lambda x+\left( {1-\lambda } \right) y} \right) \le \lambda ^{s} f\left( x \right) +\left( {1-\lambda } \right) ^{s}f\left( y \right) -c\lambda \left( {1-\lambda } \right) \left( {x-y} \right) ^2, \end{aligned}$$
for all \(x,y\in I\) and \(\lambda \in \left[ {0,1} \right] .\)
Observation 2
It is clear that, any strongly s-convex function is a strong convex function but the converse is not true in general.
Now we prove the following lemma:
Lemma 3
Let
\(f: I = \left[ {a,b} \right] \subset R\rightarrow R\)
be such that
\({f}'\)
is absolutely continuous and
\({f}''\in L_1 \left( {\left[ {a,b} \right] } \right) .\)
Then the following inequality holds:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \le \frac{\left( {b-a} \right) ^2}{96} \nonumber \\&\quad \times \int \limits _0^1 {\psi \left( {1-\psi } \right) \left[ {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) +{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) +{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right] } d\psi . \end{aligned}$$
(5)
Proof
Using integrating by parts, we have
$$\begin{aligned}&\int \limits _0^1 {\psi \left( {1-\psi } \right) {f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) d\psi } \\&\quad =-\frac{4}{b-a}\left[ {\psi \left( {1-\psi } \right) \left. {{f}'\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| _0^1 -\int \limits _0^1 {\left( {1-2\psi } \right) {f}'\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) d\psi } } \right] \\&\quad =-\frac{16}{\left( {b-a} \right) ^2}\left[ {\left( {1-2\psi } \right) \left. {f\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| _0^1 +2\int \limits _0^1 {f\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) d\psi } } \right] \\&\quad =\frac{16}{\left( {b-a} \right) ^2}\left[ {f\left( a \right) +f\left( {\frac{3a+b}{4}} \right) -\frac{96}{\left( {b-a} \right) ^3}\int \limits _a^{{\left( {3a+b} \right) } /4} {f\left( x \right) dx} } \right] . \\ \end{aligned}$$
Analogously,
$$\begin{aligned}&\int \limits _0^1 {\psi \left( {1-\psi } \right) {f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) d\psi } \\&\quad =\frac{16}{\left( {b-a} \right) ^2}\left[ {f\left( {\frac{3a+b}{4}} \right) +f\left( {\frac{a+3b}{4}} \right) } \right] -\frac{96}{\left( {b-a} \right) ^3}\int \limits _{{\left( {3a+b} \right) } / 4}^{{\left( {a+3b} \right) }/4} {f\left( x \right) dx}. \\ \end{aligned}$$
And
$$\begin{aligned}&\int \limits _0^1 {\psi \left( {1-\psi } \right) {f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) d\psi } \\&\quad =\frac{16}{\left( {b-a} \right) ^2}\left[ {f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{96}{\left( {b-a} \right) ^3}\int \limits _{{\left( {a+3b} \right) } /4}^b {f\left( x \right) dx}. \end{aligned}$$
This proves as required.
Theorem 4
Let
\(f: I = \left[ {a,b} \right] \subset R\rightarrow R\)
be such that
\({f}'\)
is absolutely continuous and
\({f}''\in L_1 \left( {\left[ {a,b} \right] } \right) .\)
If the mapping
\(\left| {{f}''} \right|\)
is strongly s-convex on
\(\left[ {a,b} \right] ,\)
for
\(q\ge 1\)
and for some fixed
\(s\in \left( {0,1} \right]\)
, then we have the following inequality:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{6^{1 / q}\left( {b-a} \right) ^2}{576}\left\{ {\begin{array}{l} \left[ {\begin{array}{l} \frac{\left( {s -5} \right) 4^{s +2}+\left( {s +9} \right) 3^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{1}{\left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q \\ -\frac{17c\left( {b-a} \right) ^2}{960} \\ \end{array}} \right] ^{1 /q} \\ +\left[ {\begin{array}{l} \frac{\left( {s -1} \right) 2^{s +2}+s +5}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{\left( {s -3} \right) 3^{s +2}+\left( {s +7} \right) 2^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q \\ -\frac{37c\left( {b-a} \right) ^2}{960} \\ \end{array}} \right] ^{1 / q} \\ +\left[ {\frac{1}{\left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{\left( {s -5} \right) 4^{s +2}+\left( {s +9} \right) 3^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q-\frac{17c\left( {b-a} \right) ^2}{960}} \right] ^{1/q} \end{array}} \right\} .\\ \end{aligned}$$
Proof
Using Lemma 3 and strongly s-convexity of \(\left| {{f}''} \right| ^q\), we have
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\begin{array}{l} \int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| } d\psi \\ +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) } \right| } d\psi +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right| } d\psi \\ \end{array}} \right] \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\int \limits _0^1 {\psi \left( {1-\psi } \right) } d\psi } \right] ^{1-1 / q}\left\{ {\begin{array}{l} \left[ {\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {\begin{array}{l} \left( {\frac{3+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{1-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1-\psi } \right) ^2\left( {3+\psi } \right) d\psi } \\ \end{array}} \right) } } \right] ^{1 /q} \\ +\left[ {\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {\begin{array}{l} \left( {\frac{1+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{3-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1-\psi ^2} \right) \left( {3-\psi } \right) d\psi } \\ \end{array}} \right) } } \right] ^{1 / q} \\ +\left[ {\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {\begin{array}{l} \left( {\frac{\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{4-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi ^2\left( {1-\psi } \right) \left( {4-\psi } \right) d\psi } \\ \end{array}} \right) } } \right] ^{1 /q} \\ \end{array}} \right\} \\&\quad =\frac{6^{1 / q}\left( {b-a} \right) ^2}{576}\left\{ {\begin{array}{l} \left[ {\begin{array}{l} \frac{\left( {s -5} \right) 4^{s +2}+\left( {s +9} \right) 3^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{1}{\left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q \\ -\frac{17c\left( {b-a} \right) ^2}{960} \\ \end{array}} \right] ^{1 /q} \\ +\left[ {\begin{array}{l} \frac{\left( {s -1} \right) 2^{s +2}+s +5}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{\left( {s -3} \right) 3^{s +2}+\left( {s +7} \right) 2^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q \\ -\frac{37c\left( {b-a} \right) ^2}{960} \\ \end{array}} \right] ^{1 /q} \\ +\left[ {\frac{1}{\left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{\left( {s -5} \right) 4^{s +2}+\left( {s +9} \right) 3^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q-\frac{17c\left( {b-a} \right) ^2}{960}} \right] ^{1 / q} \\ \end{array}} \right\} . \end{aligned}$$
Corollary 5
Under the conditions of Theorem
4,
-
1.
If \(q = 1\), then
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {s -3} \right) 4^{s +2}+\left( {2s +6} \right) \left( {3^{s +2}+1} \right) }{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left( {\left| {{f}''\left( a \right) } \right| +\left| {{f}''\left( b \right) } \right| } \right) -\frac{71c\left( {b-a} \right) ^2}{960}} \right] . \end{aligned}$$
-
2.
If \(q = 1\) and \(s = 1\), then
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\left( {\left| {{f}''\left( a \right) } \right| +\left| {{f}''\left( b \right) } \right| } \right) -\frac{71c\left( {b-a} \right) ^2}{960}} \right] .
\end{aligned}$$
Theorem 6
Let
f
be defined as in Theorem 4 and the mapping \(\left| {{f}''} \right| ^q\)
is strongly s-convex on
\(\left[ {a,b} \right] ,\)
for
\(q>1\)
and for some fixed
\(s\in \left( {0,1} \right]\)
, then we have the following inequality:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {B\left( {\frac{2q-1}{q-1},\frac{2q-1}{q-1}} \right) } \right] ^{1-1 /q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) }} \right) } \right] ^{1/q} \\&\qquad \times \, \left\{ {\begin{array}{l} \left[ {\left( {4^{s +1}-3^{s +1}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left| {{f}''\left( b \right) } \right| ^q-\frac{5c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1/ q} \\ +\left[ {\left( {2^{s +1}-1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +1}-2^{s +1}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{11c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1 /q} \\ +\left[ {\left| {{f}''\left( a \right) } \right| ^q+\left( {4^{s +1}-3^{s +1}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{5c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1 /q} \\ \end{array}} \right\} , \end{aligned}$$
where \(B\left( {\alpha ,\beta } \right)\) is the classical Beta function which may be defined by
$$\begin{aligned} B\left( {\alpha ,\beta } \right) =\int \limits _0^1 {\psi ^{\alpha -1}\left( {1-\psi } \right) ^{\beta -1}} d\psi ,\quad s ,\beta >0. \end{aligned}$$
Proof
Using Lemma 3, strong s-convexity of \(\left| {{f}''} \right| ^q\) and Holder’s inequality, we have
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\begin{array}{l} \int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| } d\psi \\ +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) } \right| } d\psi +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right| } d\psi \\ \end{array}} \right] \\&\quad \begin{array}{l} \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\int \limits _0^1 {\left[ {\psi \left( {1-\psi } \right) } \right] } ^{q / {\left( {q-1} \right) }}d\psi } \right] ^{1-1 / q}\left\{ {\begin{array}{l} \left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {\left( {\frac{3+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{1-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\left( {1-\psi } \right) \left( {3+\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 /q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {\left( {\frac{1+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{3-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\left( {1-\psi } \right) \left( {3-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 /q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {\left( {\frac{\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{4-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {4-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1/q} \\ \end{array}} \right\} \\ \end{array}\\&\qquad \times \begin{array}{l} \left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\ \le \frac{\left( {b-a} \right) ^2}{96}\left[ {B\left( {\frac{2q-1}{q-1},\frac{2q-1}{q-1}} \right) } \right] ^{1-1/q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) }} \right) } \right] ^{1/q} \\ \end{array}\\&\qquad \times \, \left\{ {\begin{array}{l} \left[ {\left( {4^{s +1}-3^{s +1}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left| {{f}''\left( b \right) } \right| ^q-\frac{5c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1/q} \\ +\left[ {\left( {2^{s +1}-1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +1}-2^{s +1}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{11c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1 /q} \\ +\left[ {\left| {{f}''\left( a \right) } \right| ^q+\left( {4^{s +1}-3^{s +1}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{5c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1 /q} \end{array}} \right\} . \end{aligned}$$
This completes the proof.
Theorem 7
Let
f
be defined as in Theorem
4
and the mapping
\(\left| {{f}''} \right| ^q\)
is strongly s-convex on
\(\left[ {a,b} \right] ,\)
for
\(q>1\)
and for some fixed
\(s\in \left( {0,1} \right]\)
, then we have the following inequality:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {q-1} \right) ^2}{\left( {2q-1} \right) \left( {3q-2} \right) }} \right] ^{1-1 / q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) \left( {s +2} \right) }} \right) } \right] ^{1/ q} \\&\qquad \times \,\left\{ {\begin{array}{l} \left[ {\left( {\left( {s -2} \right) 4^{s +1}+3^{s +2}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left| {{f}''\left( b \right) } \right| ^q-\frac{7c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ +\left[ {\left( {2^{s +1}s +1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +2}-2^{s +1}\left( {s +4} \right) } \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{23c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 / q} \\ +\left[ {\left( {s +1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {4^{s +2}-3^{s +1}\left( {s +5} \right) } \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{13c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ \end{array}} \right\} . \\ \end{aligned}$$
Proof
Using Lemma 3, Holder’s inequality and strongly s- convexity of \(\left| {{f}''} \right| ^q\), we have
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\begin{array}{l} \int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| } d\psi \\ +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) } \right| } d\psi +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right| } d\psi \\ \end{array}} \right] \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\int \limits _0^1 {\left[ {\psi \left( {1-\psi } \right) } \right] } ^{q / {\left( {q-1} \right) }}d\psi } \right] ^{1-1/ q}\left\{ {\begin{array}{l} \left[ {\left( {\begin{array}{l} \int \limits _0^1 {\psi \left( {\left( {\frac{3+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{1-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {3+\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\psi \left( {\left( {\frac{1+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{3-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1+\psi } \right) \left( {3-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\psi \left( {\left( {\frac{\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{4-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi ^2\left( {4-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ \end{array}} \right\} . \\ \end{aligned}$$
or
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {q-1} \right) ^2}{\left( {2q-1} \right) \left( {3q-2} \right) }} \right] ^{1-1 / q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) \left( {s +2} \right) }} \right) } \right] ^{1/q} \\&\quad \times \, \left\{ {\begin{array}{l} \left[ {\left( {\left( {s -2} \right) 4^{s +1}+3^{s +2}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left| {{f}''\left( b \right) } \right| ^q-\frac{7c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 / q} \\ +\left[ {\left( {2^{s +1}s +1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +2}-2^{s +1}\left( {s +4} \right) } \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{23c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ +\left[ {\left( {s +1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {4^{s +2}-3^{s +1}\left( {s +5} \right) } \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{13c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 / q} \\ \end{array}} \right\} \end{aligned}$$
This completes the proof.
Theorem 8
Let
f
be defined as in Theorem 4
and the mapping
\(\left| {{f}''} \right| ^q\)
is strongly s-convex on
\(\left[ {a,b} \right] ,\)
for
\(q>1\)
and for some fixed
\(s\in \left( {0,1} \right]\)
, then we have the following inequality:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {q-1} \right) ^2}{\left( {2q-1} \right) \left( {3q-2} \right) }} \right] ^{1-1/ q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) \left( {s +2} \right) }} \right) } \right] ^{1 /q} \\&\qquad \times \, \left\{ {\begin{array}{l} \left[ {\left( {4^{s +2}-\left( {s +5} \right) 3^{s +1}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {s +1} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{13c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/q} \\ +\left[ {\left( {2^{s +2}-s -3} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +1}\left( {s -1} \right) -2^{s +2}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{21c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/q} \\ +\left[ {\left| {{f}''\left( a \right) } \right| ^q+\left( {\left( {s -2} \right) 4^{s +1}+3^{s +2}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{7c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ \end{array}} \right\} . \end{aligned}$$
Proof
Using Lemma 3, Holder inequality and strongly s- convexity of \(\left| {{f}''} \right|\), we have
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\begin{array}{l} \int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| } d\psi \\ +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) } \right| } d\psi +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right| } d\psi \\ \end{array}} \right] \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\int \limits _0^1 {\left( {1-\psi } \right) } \psi ^{q /{\left( {q-1} \right) }}d\psi } \right] ^{1-1 / q}\left\{ {\begin{array}{l} \left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {1-\psi } \right) \left( {\left( {\frac{3+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{1-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\left( {1-\psi } \right) ^2\left( {3+\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {1-\psi } \right) \left( {\left( {\frac{1+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{3-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\left( {1-\psi ^2} \right) \left( {3-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {1-\psi } \right) \left( {\left( {\frac{\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{4-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {4-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ \end{array}} \right\} . \\ \end{aligned}$$
or
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {q-1} \right) ^2}{\left( {2q-1} \right) \left( {3q-2} \right) }} \right] ^{1-1/ q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) \left( {s +2} \right) }} \right) } \right] ^{1/ q} \\&\qquad \times \, \left\{ {\begin{array}{l} \left[ {\left( {4^{s +2}-\left( {s +5} \right) 3^{s +1}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {s +1} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{13c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 / q} \\ +\left[ {\left( {2^{s +2}-s -3} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +1}\left( {s -1} \right) -2^{s +2}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{21c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ +\left[ {\left| {{f}''\left( a \right) } \right| ^q+\left( {\left( {s -2} \right) 4^{s +1}+3^{s +2}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{7c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 /q} \\ \end{array}} \right\} . \end{aligned}$$
This completes the proof. \(\square\)