1 Introduction
It is a well known fact that inequalities have important role in the studies of inequality theory, linear programming, extremum problems, optimization, error estimates and game theory. Over the years, only integer real order integrals were taken into account while handling new results about integral inequalities. However, in the recent years fractional integral operator have been considered by many scientists (see [1 ],[2 ],[3 ],[4 ],[5 ],[6 ],[7 ],[8 ],[9 ],[10 ],[11 ],[12 ]) and the references therein. There are some inequalities in the literature that accelerates studies on integral inequalities. In the following part, the Hermite-Hadamard inequality which is one of the most famous and practical inequality in the literature is given:
Theorem 1
Let f be defined from interval I (a nonempty subset of ℝ) to ℝ be a convex function on I and m, n ∈ I with m < n. Then the double inequality given in the following holds:
(1)
f ( m + n 2 ) ≤ 1 n − m ∫ m n f ( x ) dx ≤ f ( m ) + f ( n ) 2 .
f\left({{{m + n} \over 2}} \right) \le {1 \over {n - m}}\int_m^n f\left(x \right)dx \le {{f\left(m \right) + f\left(n \right)} \over 2}.
Now we will mention about Riemann-Liouville fractional integration operator (see [6 ]) which ables to integrate functions on fractional orders.
Definition 1
Let f ∈ L 1 [m , n ].
J m + α f
J_{m +}^\alpha f
and
J n − α f
J_{n -}^\alpha f
which are called left-sided and right-sided Riemann-Liouville integrals of order α > 0 with 0 ≤ m ≤ x ≤ n are defined by
(2)
J m + α f = 1 Γ ( α ) ∫ m x ( x − t ) α − 1 f ( t ) dt , x > m
J_{m +}^\alpha f = {1 \over {\Gamma \left(\alpha \right)}}\int_m^x {\left({x - t} \right)^{\alpha - 1}}f\left(t \right)dt{\rm{,}}\;\;\;x > m
and
(3)
J n − α f = 1 Γ ( α ) ∫ x n ( t − x ) α − 1 f ( t ) dt , x < n
J_{n -}^\alpha f = {1 \over {\Gamma \left(\alpha \right)}}\int_x^n {\left({t - x} \right)^{\alpha - 1}}f\left(t \right)dt{\rm{,}}\;\;\;x < n
respectively where
Γ ( α ) = ∫ 0 ∞ e − t u α − 1 du
\Gamma \left(\alpha \right) = \int_0^\infty {e^{- t}}{u^{\alpha - 1}}du
. Here
J m + 0 f ( x ) = J n − 0 f ( x ) = f ( x )
J_{{m^ +}}^0f\left(x \right) = J_{{n^ -}}^0f\left(x \right) = f\left(x \right)
.
The results have been put forward inspiring from the following kernel obtained in [9 ].
Lemma 2
Let f : I ⊆ ℝ → ℝ be a differentiable mapping on I° and m, n ∈ I with m < n . If f ′ ∈ L 1 ([m , n ]), λ, μ ∈ ℝ, and ξ ∈ [0, 1], then
λ f ( m ) + μ f ( n ) 2 + ( 2 − λ − μ ) 2 f ( ξ m + ( 1 − ξ ) n ) − 1 n − m ∫ m n f ( x ) dx = n − m 2 [ ( 1 − ξ ) ∫ 0 1 ( 2 ( 1 − ξ ) t − λ ) f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) m ) dt + ξ ∫ 0 1 ( μ − 2 ξ t ) f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) n ) dt ] .
\matrix{ {{\kern 10pt }{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)} \over 2}f\left({\xi m + \left({1 - \xi} \right)n} \right) - {1 \over {n - m}}\int_m^n f\left(x \right)dx} \hfill \cr = {{{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left({2\left({1 - \xi} \right)t - \lambda} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \right.} \hfill \cr {\kern 10pt }{\left. {+ \xi \int_0^1 \left({\mu - 2\xi t} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \right].} \hfill \cr}
Now firstly, we will give a new lemma including Riemann-Liouville fractional integral operator, then we will obtain new inequalities for convex functions.
2 Results Via Riemann-Liouville Fractional Integrals
Lemma 3
Let f : I ⊆ ℝ → ℝ be a differentiable mapping on I° and m, n ∈ I with m < n. If f ′ ∈ L 1 ([m , n ]), λ, μ ∈ ℝ, α > 0 and ξ ∈ [0, 1] then
λ f ( m ) + μ f ( n ) 2 + ( 2 − λ − μ ) 2 f ( ξ m + ( 1 − ξ ) n ) − Γ ( α + 1 ) ( n − m ) α [ ( 1 − ξ ) 1 − α J [ ξ m + ( 1 − ξ ) n ] − α f ( m ) + ξ 1 − α J [ ξ m + ( 1 − ξ ) n ] + α f ( n ) ] = n − m 2 [ ( 1 − ξ ) ∫ 0 1 ( 2 ( 1 − ξ ) t α − λ ) f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) m ) dt + ξ ∫ 0 1 ( μ − 2 ξ t α ) f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) n ) dt ]
\matrix{{\kern 10pt}{{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)} \over 2}f\left({\xi m + \left({1 - \xi} \right)n} \right)} \hfill \cr {\kern 10pt} {- {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \hfill \cr = {{{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left({2\left({1 - \xi} \right){t^\alpha} - \lambda} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \right.} \hfill \cr {\kern 10pt} {\left. {+ \xi \int_0^1 \left({\mu - 2\xi {t^\alpha}} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \right]} \hfill}
where Γ(.) is the gamma function .
Proof
Integrating by part and changing variables of integration x = t (ξ m + (1 − ξ ) n ) + (1 − t ) m yield
(4)
∫ 0 1 ( 2 ( 1 − ξ ) t α − λ ) f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) m ) dt = ( 2 ( 1 − ξ ) t α − λ ) f ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) m ) ( 1 − ξ ) ( n − m ) | 0 1 − 2 α ( n − m ) ∫ 0 1 t α − 1 f ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) m ) dt = [ 2 ( 1 − ξ ) − λ ] f ( ξ m + ( 1 − ξ ) n ) + λ f ( m ) ( 1 − ξ ) ( n − m ) − 2 α ( n − m ) ∫ 0 1 t α − 1 f ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) m ) dt = [ 2 ( 1 − ξ ) − λ ] f ( ξ m + ( 1 − ξ ) n ) + λ f ( m ) ( 1 − ξ ) ( n − m ) − 2 α ( 1 − ξ ) α ( n − m ) α + 1 ∫ m ξ m + ( 1 − ξ ) n ( x − m ) α − 1 f ( x ) dx = [ 2 ( 1 − ξ ) − λ ] f ( ξ m + ( 1 − ξ ) n ) + λ f ( m ) ( 1 − ξ ) ( n − m ) − 2 Γ ( α + 1 ) ( 1 − ξ ) α ( n − m ) α + 1 J [ ξ m + ( 1 − ξ ) n ] − α f ( m ) .
\matrix{{\,\,\,\,\,\int_0^1 \left({2\left({1 - \xi} \right){t^\alpha} - \lambda} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \hfill \cr {= \left. {\left({2\left({1 - \xi} \right){t^\alpha} - \lambda} \right){{f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)} \over {\left({1 - \xi} \right)\left({n - m} \right)}}} \right|_0^1} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {\left({n - m} \right)}}\int_0^1 {t^{\alpha - 1}}f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \hfill \cr {= {{\left[ {2\left({1 - \xi} \right) - \lambda} \right]f\left({\xi m + \left({1 - \xi} \right)n} \right) + \lambda f\left(m \right)} \over {\left({1 - \xi} \right)\left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {\left({n - m} \right)}}\int_0^1 {t^{\alpha - 1}}f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)dt} \hfill \cr {= {{\left[ {2\left({1 - \xi} \right) - \lambda} \right]f\left({\xi m + \left({1 - \xi} \right)n} \right) + \lambda f\left(m \right)} \over {\left({1 - \xi} \right)\left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {{{\left({1 - \xi} \right)}^\alpha}{{\left({n - m} \right)}^{\alpha + 1}}}}\int_m^{\xi m + \left({1 - \xi} \right)n} {{\left({x - m} \right)}^{\alpha - 1}}f\left(x \right)dx} \hfill \cr {= {{\left[ {2\left({1 - \xi} \right) - \lambda} \right]f\left({\xi m + \left({1 - \xi} \right)n} \right) + \lambda f\left(m \right)} \over {\left({1 - \xi} \right)\left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\Gamma \left({\alpha + 1} \right)} \over {{{\left({1 - \xi} \right)}^\alpha}{{\left({n - m} \right)}^{\alpha + 1}}}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right).} \hfill \cr}
On the other hand, with similar way and changing variables of integration x = t (ξ m + (1 − ξ ) n )+(1 − t ) n yield
(5)
∫ 0 1 ( μ − 2 ξ t α ) f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) n ) dt = ( μ − 2 ξ t α ) f ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) n ) ξ ( m − n ) | 0 1 − 2 α ( n − m ) ∫ 0 1 t α − 1 f ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) n ) dt = ( 2 ξ − μ ) f ( ξ m + ( 1 − ξ ) n ) + μ f ( n ) ξ ( n − m ) − 2 α ( n − m ) ∫ 0 1 t α − 1 f ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) n ) dt = ( 2 ξ − μ ) f ( ξ m + ( 1 − ξ ) n ) + μ f ( n ) ξ ( n − m ) − 2 α ξ α ( n − m ) α + 1 ∫ ξ m + ( 1 − ξ ) n n ( n − x ) α − 1 f ( x ) dx = ( 2 ξ − μ ) f ( ξ m + ( 1 − ξ ) n ) + μ f ( n ) ξ ( n − m ) − 2 Γ ( α + 1 ) ξ α ( n − m ) α + 1 J [ ξ m + ( 1 − ξ ) n ] + α f ( n ) .
\matrix{{\,\,\,\,\,\int_0^1 \left({\mu - 2\xi {t^\alpha}} \right){f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \hfill \cr {= \left. {\left({\mu - 2\xi {t^\alpha}} \right){{f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)} \over {\xi \left({m - n} \right)}}} \right|_0^1} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {\left({n - m} \right)}}\int_0^1 {t^{\alpha - 1}}f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \hfill \cr {= {{\left({2\xi - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right) + \mu f\left(n \right)} \over {\xi \left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {\left({n - m} \right)}}\int_0^1 {t^{\alpha - 1}}f\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)dt} \hfill \cr {= {{\left({2\xi - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right) + \mu f\left(n \right)} \over {\xi \left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\alpha} \over {{\xi ^\alpha}{{\left({n - m} \right)}^{\alpha + 1}}}}\int_{\xi m + \left({1 - \xi} \right)n}^n {{\left({n - x} \right)}^{\alpha - 1}}f\left(x \right)dx} \hfill \cr {= {{\left({2\xi - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right) + \mu f\left(n \right)} \over {\xi \left({n - m} \right)}}} \hfill \cr {\,\,\,\,\, - {{2\Gamma \left({\alpha + 1} \right)} \over {{\xi ^\alpha}{{\left({n - m} \right)}^{\alpha + 1}}}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right).} \hfill \cr}
By multiplying 4 with (1 − ξ ), 5 with ξ , summing them side by side and multiplying the equality with
n − m 2
{{n - m} \over 2}
we get the desired result.
Remark 1
If we choose α = 1 in Lemma 3, we gel Lemma 2.
Theorem 4
Let f : I ⊆ ℝ → ℝ be a differentiable mapping on I° and m, n ∈ I with m < n . If |f ′| is convex on I, f′ ∈ L 1 ([m , n ]), λ, μ ∈ ℝ+ , α > 0 and ξ ∈ [0, 1] then
| λ f ( m ) + μ f ( n ) 2 + ( 2 − λ − μ ) f ( ξ m + ( 1 − ξ ) n ) 2 − Γ ( α + 1 ) ( n − m ) α [ ( 1 − ξ ) 1 − α J [ ξ m + ( 1 − ξ ) n ] − α f ( m ) + ξ 1 − α J [ ξ m + ( 1 − ξ ) n ] + α f ( n ) ] | ≤ n − m 2 { ( 1 − ξ ) [ ρ 1 | f ' ( ξ m + ( 1 − ξ ) n ) | + Ψ 1 | f ' ( m ) | ] + ξ [ γ 1 | f ' ( ξ m + ( 1 − ξ ) n ) | + δ 1 | f ' ( m ) | ] for λ < 2 ( 1 − ξ ) and μ < 2 ξ ( 1 − ξ ) [ ρ 1 | f ' ( ξ m + ( 1 − ξ ) n ) | + Ψ 1 | f ' ( m ) | ] + ξ [ γ 2 | f ' ( ξ m + ( 1 − ξ ) n ) | + δ 2 | f ' ( m ) | ] for λ < 2 ( 1 − ξ ) and μ ≥ 2 ξ ( 1 − ξ ) [ ρ 2 | f ' ( ξ m + ( 1 − ξ ) n ) | + Ψ 2 | f ' ( m ) | ] + ξ [ γ 1 | f ' ( ξ m + ( 1 − ξ ) n ) | + δ 1 | f ' ( m ) | ] for λ ≥ 2 ( 1 − ξ ) and μ < 2 ξ ( 1 − ξ ) [ ρ 2 | f ' ( ξ m + ( 1 − ξ ) n ) | + Ψ 2 | f ' ( m ) | ] + ξ [ γ 2 | f ' ( ξ m + ( 1 − ξ ) n ) | + δ 2 | f ' ( m ) | ] for λ ≥ 2 ( 1 − ξ ) and μ ≥ 2 ξ
\matrix{{\,\,\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left\{{\matrix{{\matrix{{\left({1 - \xi} \right)\left[ {{\rho _1}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\Psi _1}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr {+ \xi \left[ {{\gamma _1}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\delta _1}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr}} & {for\,\,\,\lambda < 2\left({1 - \xi} \right)\,\,\,and\,\,\,\mu < 2\xi} \cr {} & {} \cr {\matrix{{\left({1 - \xi} \right)\left[ {{\rho _1}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\Psi _1}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr {+ \xi \left[ {{\gamma _2}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\delta _2}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr}} & {for\,\,\,\lambda < 2\left({1 - \xi} \right)\,\,\,and\,\,\,\mu \ge 2\xi} \cr {} & {} \cr {\matrix{{\left({1 - \xi} \right)\left[ {{\rho _2}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\Psi _2}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr {+ \xi \left[ {{\gamma _1}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\delta _1}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr}} & {for\,\,\,\lambda \ge 2\left({1 - \xi} \right)\,\,\,and\,\,\,\mu < 2\xi} \cr {} & {} \cr {\matrix{{\left({1 - \xi} \right)\left[ {{\rho _2}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\Psi _2}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr {+ \xi \left[ {{\gamma _2}\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + {\delta _2}\left| {{f^{'}}\left(m \right)} \right|} \right]} \cr}} & {for\,\,\,\lambda \ge 2\left({1 - \xi} \right)\,\,\,and\,\,\,\mu \ge 2\xi} \cr}} \right.} \hfill \cr}
where Γ(.) is the gamma function and
ρ 1 = λ 2 + α α 2 2 α ( 1 − ξ ) 2 α − λ 2 + α α 2 2 − α α ( 1 − ξ ) 2 α ( 2 + α ) + 2 ( 1 − ξ ) 2 + α − λ 2 ρ 2 = λ 2 − 2 ( 1 − ξ ) 2 + α Ψ 1 = λ 1 + α α 2 1 − α α ( 1 − ξ ) 1 α − λ 1 + α α 2 1 − α α ( 1 + α ) ( 1 − ξ ) 1 α + 2 ( 1 − ξ ) 1 + α − λ − ρ 1 Ψ 2 = λ 2 − 2 ( 1 − ξ ) 1 + α + 2 ( 1 − ξ ) 2 + α γ 1 = μ 2 + α α 2 2 α ξ 2 α − μ 2 + α α 2 2 − α α ( 2 + α ) ξ 2 α + 2 ξ 2 + α − μ 2 γ 2 = μ 2 − 2 ξ 2 + α δ 1 = μ 1 + α α 2 1 − α α ξ 1 α − μ 1 + α α 2 1 − α α ( 1 + α ) ξ 1 α + 2 ξ 1 + α − μ − γ 1 δ 2 = μ 2 − 2 ξ 1 + α + 2 ξ 2 + α .
\matrix{{{\rho _1} = {{{\lambda ^{{{2 + \alpha} \over \alpha}}}} \over {{2^{{2 \over \alpha}}}{{\left({1 - \xi} \right)}^{{2 \over \alpha}}}}} - {{{\lambda ^{{{2 + \alpha} \over \alpha}}}} \over {{2^{{{2 - \alpha} \over \alpha}}}{{\left({1 - \xi} \right)}^{{2 \over \alpha}}}\left({2 + \alpha} \right)}} + {{2\left({1 - \xi} \right)} \over {2 + \alpha}} - {\lambda \over 2}} \hfill \cr {{\rho _2} = {\lambda \over 2} - {{2\left({1 - \xi} \right)} \over {2 + \alpha}}} \hfill \cr {{\Psi _1} = {{{\lambda ^{{{1 + \alpha} \over \alpha}}}} \over {{2^{{{1 - \alpha} \over \alpha}}}{{\left({1 - \xi} \right)}^{{1 \over \alpha}}}}} - {{{\lambda ^{{{1 + \alpha} \over \alpha}}}} \over {{2^{{{1 - \alpha} \over \alpha}}}\left({1 + \alpha} \right){{\left({1 - \xi} \right)}^{{1 \over \alpha}}}}} + {{2\left({1 - \xi} \right)} \over {1 + \alpha}} - \lambda - {\rho _1}} \hfill \cr {{\Psi _2} = {\lambda \over 2} - {{2\left({1 - \xi} \right)} \over {1 + \alpha}} + {{2\left({1 - \xi} \right)} \over {2 + \alpha}}} \hfill \cr {{\gamma _1} = {{{\mu ^{{{2 + \alpha} \over \alpha}}}} \over {{2^{{2 \over \alpha}}}{\xi ^{{2 \over \alpha}}}}} - {{{\mu ^{{{2 + \alpha} \over \alpha}}}} \over {{2^{{{2 - \alpha} \over \alpha}}}\left({2 + \alpha} \right){\xi ^{{2 \over \alpha}}}}} + {{2\xi} \over {2 + \alpha}} - {\mu \over 2}} \hfill \cr {{\gamma _2} = {\mu \over 2} - {{2\xi} \over {2 + \alpha}}} \hfill \cr {{\delta _1} = {{{\mu ^{{{1 + \alpha} \over \alpha}}}} \over {{2^{{{1 - \alpha} \over \alpha}}}{\xi ^{{1 \over \alpha}}}}} - {{{\mu ^{{{1 + \alpha} \over \alpha}}}} \over {{2^{{{1 - \alpha} \over \alpha}}}\left({1 + \alpha} \right){\xi ^{{1 \over \alpha}}}}} + {{2\xi} \over {1 + \alpha}} - \mu - {\gamma _1}} \hfill \cr {{\delta _2} = {\mu \over 2} - {{2\xi} \over {1 + \alpha}} + {{2\xi} \over {2 + \alpha}}.} \hfill \cr}
Proof
By using Lemma 3 and using properties of absolute value we have
| λ f ( m ) + μ f ( n ) 2 + ( 2 − λ − μ ) f ( ξ m + ( 1 − ξ ) n ) 2 − Γ ( α + 1 ) ( n − m ) α [ ( 1 − ξ ) 1 − α J [ ξ m + ( 1 − ξ ) n ] − α f ( m ) + ξ 1 − α J [ ξ m + ( 1 − ξ ) n ] + α f ( n ) ] | ≤ n − m 2 [ ( 1 − ξ ) ∫ 0 1 | 2 ( 1 − ξ ) t α − λ | | f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) m ) | dt + ξ ∫ 0 1 | μ − 2 ξ t α | | f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) n ) | dt ] .
\matrix{{\,\,\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)} \right|dt} \right.} \hfill \cr {\left. {\,\,\,\,\, + \xi \int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)} \right|dt} \right].} \hfill \cr}
Using convexity of |f ′| we have
(6)
| λ f ( m ) + μ f ( n ) 2 + ( 2 − λ − μ ) f ( ξ m + ( 1 − ξ ) n ) 2 − Γ ( α + 1 ) ( n − m ) α [ ( 1 − ξ ) 1 − α J [ ξ m + ( 1 − ξ ) n ] − α f ( m ) + ξ 1 − α J [ ξ m + ( 1 − ξ ) n ] + α f ( n ) ] | ≤ n − m 2 [ ( 1 − ξ ) ∫ 0 1 | 2 ( 1 − ξ ) t α − λ | [ t | f ' ( ξ m + ( 1 − ξ ) n ) | + ( 1 − t ) | f ' ( m ) | ] dt + ξ ∫ 0 1 | μ − 2 ξ t α | [ t | f ' ( ξ m + ( 1 − ξ ) n ) | + ( 1 − t ) | f ' ( n ) | ] dt ] .
\matrix{{\,\,\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|\left[ {t\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + \left({1 - t} \right)\left| {{f^{'}}\left(m \right)} \right|} \right]dt} \right.} \hfill \cr {\left. {\,\,\,\, + \xi \int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|\left[ {t\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right| + \left({1 - t} \right)\left| {{f^{'}}\left(n \right)} \right|} \right]dt} \right].} \hfill \cr}
With simple calculations it can be seen
∫ 0 1 t | 2 ( 1 − ξ ) t α − λ | dt = { ρ 1 , λ < 2 ( 1 − ξ ) ρ 2 , λ ≥ 2 ( 1 − ξ ) ∫ 0 1 ( 1 − t ) | 2 ( 1 − ξ ) t α − λ | dt = { Ψ 1 , λ < 2 ( 1 − ξ ) Ψ 2 , λ ≥ 2 ( 1 − ξ ) ∫ 0 1 t | μ − 2 ξ t α | dt = { γ 1 , μ < 2 ξ γ 2 , μ ≥ 2 ξ ∫ 0 1 ( 1 − t ) | μ − 2 ξ t α | dt = { δ 1 , μ < 2 ξ δ 2 , μ ≥ 2 ξ .
\matrix{{\kern 32pt}{\int_0^1 t\left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt =} \hfill & {\left\{{\matrix{{{\rho _1},} & {\lambda < 2\left({1 - \xi} \right)} \cr {{\rho _2},} & {\lambda \ge 2\left({1 - \xi} \right)} \cr}} \right.} \hfill \cr {\int_0^1 \left({1 - t} \right)\left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt =} \hfill & {\left\{{\matrix{{{\Psi _1},} & {\lambda < 2\left({1 - \xi} \right)} \cr {{\Psi _2},} & {\lambda \ge 2\left({1 - \xi} \right)} \cr}} \right.} \hfill \cr {\kern 65pt}{\int_0^1 t\left| {\mu - 2\xi {t^\alpha}} \right|dt =} \hfill & {\left\{{\matrix{{{\gamma _1},} & {\mu < 2\xi} \cr {{\gamma _2},} & {\mu \ge 2\xi} \cr}} \right.} \hfill \cr{\kern 32pt} {\int_0^1 \left({1 - t} \right)\left| {\mu - 2\xi {t^\alpha}} \right|dt =} \hfill & {\left\{{\matrix{{{\delta _1},} & {\mu < 2\xi} \cr {{\delta _2},} & {\mu \ge 2\xi} \cr}} \right..} \hfill \cr}
By using necessary coefficients in (6) , the proof is completed.
Theorem 5
Let f : I ⊆ ℝ → ℝ be a differentiable mapping on I° and m, n ∈ I with m < n . If |f ′|q is convex on I, f ′ ∈ L 1 ([m , n ]), λ , μ ∈ ℝ+ , α > 0, ξ ∈ [0, 1] then
| λ f ( m ) + μ f ( n ) 2 + ( 2 − λ − μ ) f ( ξ m + ( 1 − ξ ) n ) 2 − Γ ( α + 1 ) ( n − m ) α [ ( 1 − ξ ) 1 − α J [ ξ m + ( 1 − ξ ) n ] − α f ( m ) + ξ 1 − α J [ ξ m + ( 1 − ξ ) n ] + α f ( n ) ] | ≤ n − m 2 { ( 1 − ξ ) ( ζ 1 ) 1 − 1 q [ ρ 1 | f ' ( ξ m + ( 1 − ξ ) n ) | q + Ψ 1 | f ' ( m ) | q ] 1 q + ξ ( η 1 ) 1 − 1 q [ γ 1 | f ' ( ξ m + ( 1 − ξ ) n ) | q + δ 1 | f ' ( n ) | q ] 1 q , for λ < 2 ( 1 − ξ ) and μ < 2 ξ ; ( 1 − ξ ) ( ζ 1 ) 1 − 1 q [ ρ 1 | f ' ( ξ m + ( 1 − ξ ) n ) | q + Ψ 1 | f ' ( m ) | q ] 1 q + ξ ( η 2 ) 1 − 1 q [ γ 2 | f ' ( ξ m + ( 1 − ξ ) n ) | q + δ 2 | f ' ( n ) | q ] 1 q , for λ < 2 ( 1 − ξ ) and μ ≥ 2 ξ ; ( 1 − ξ ) ( ζ 2 ) 1 − 1 q [ ρ 2 | f ' ( ξ m + ( 1 − ξ ) n ) | q + Ψ 2 | f ' ( m ) | q ] 1 q + ξ ( η 1 ) 1 − 1 q [ γ 1 | f ' ( ξ m + ( 1 − ξ ) n ) | q + δ 1 | f ' ( n ) | q ] 1 q , for λ ≥ 2 ( 1 − ξ ) and μ < 2 ξ ; ( 1 − ξ ) ( ζ 2 ) 1 − 1 q [ ρ 2 | f ' ( ξ m + ( 1 − ξ ) n ) | q + Ψ 2 | f ' ( m ) | q ] 1 q + ξ ( η 2 ) 1 − 1 q [ γ 2 | f ' ( ξ m + ( 1 − ξ ) n ) | q + δ 2 | f ' ( n ) | q ] 1 q , for λ ≥ 2 ( 1 − ξ ) and μ ≥ 2 ξ
\matrix{{\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {- {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left\{{\matrix{{\matrix{{\left({1 - \xi} \right){{\left({{\zeta _1}} \right)}^{1 - {1 \over q}}}{{\left[ {{\rho _1}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\Psi _1}{{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]}^{{1 \over q}}}} \cr {+ \xi {{\left({{\eta _1}} \right)}^{1 - {1 \over q}}}{{\left[ {{\gamma _1}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\delta _1}{{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]}^{{1 \over q}}},} \cr {{\rm{for}}\lambda < 2\left({1 - \xi} \right){\rm{and}}\mu < 2\xi ;} \cr}} \cr {} \cr {\matrix{{\left({1 - \xi} \right){{\left({{\zeta _1}} \right)}^{1 - {1 \over q}}}{{\left[ {{\rho _1}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\Psi _1}{{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]}^{{1 \over q}}}} \cr {+ \xi {{\left({{\eta _2}} \right)}^{1 - {1 \over q}}}{{\left[ {{\gamma _2}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\delta _2}{{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]}^{{1 \over q}}},} \cr {{\rm{for}}\lambda < 2\left({1 - \xi} \right){\rm{and}}\mu \ge 2\xi ;} \cr}} \cr {} \cr {\matrix{{\left({1 - \xi} \right){{\left({{\zeta _2}} \right)}^{1 - {1 \over q}}}{{\left[ {{\rho _2}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\Psi _2}{{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]}^{{1 \over q}}}} \cr {+ \xi {{\left({{\eta _1}} \right)}^{1 - {1 \over q}}}{{\left[ {{\gamma _1}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\delta _1}{{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]}^{{1 \over q}}},} \cr {{\rm{for}}\lambda \ge 2\left({1 - \xi} \right){\rm{and}}\mu < 2\xi ;} \cr}} \cr {} \cr {\matrix{{\left({1 - \xi} \right){{\left({{\zeta _2}} \right)}^{1 - {1 \over q}}}{{\left[ {{\rho _2}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\Psi _2}{{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]}^{{1 \over q}}}} \cr {+ \xi {{\left({{\eta _2}} \right)}^{1 - {1 \over q}}}{{\left[ {{\gamma _2}{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + {\delta _2}{{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]}^{{1 \over q}}},} \cr {{\rm{for}}\lambda \ge 2\left({1 - \xi} \right){\rm{and}}\mu \ge 2\xi} \cr}} \cr}} \right.} \hfill \cr}
where q ≥ 1, Γ(.) is the gamma function and
ζ 1 = 2 λ λ 2 ( 1 − ξ ) α − 4 ( 1 − ξ ) ( α + 1 ) ( λ 2 ( 1 − ξ ) ) α + 1 α + 2 ( 1 − ξ ) α + 1 − λ ζ 2 = λ − 2 ( 1 − ξ ) α + 1 η 1 = 2 μ μ 2 ξ α − 4 ξ ( α + 1 ) ( μ 2 ξ ) α + 1 α + 2 ξ α + 1 − μ η 2 = μ − 2 ξ α + 1
\matrix{{{\zeta _1} = 2\lambda \root \alpha \of {{\lambda \over {2\left({1 - \xi} \right)}}} - {{4\left({1 - \xi} \right)} \over {\left({\alpha + 1} \right)}}\root \alpha \of {{{\left({{\lambda \over {2\left({1 - \xi} \right)}}} \right)}^{\alpha + 1}}} + {{2\left({1 - \xi} \right)} \over {\alpha + 1}} - \lambda} \hfill \cr {{\zeta _2} = \lambda - {{2\left({1 - \xi} \right)} \over {\alpha + 1}}} \hfill \cr {{\eta _1} = 2\mu \root \alpha \of {{\mu \over {2\xi}}} - {{4\xi} \over {\left({\alpha + 1} \right)}}\root \alpha \of {{{\left({{\mu \over {2\xi}}} \right)}^{\alpha + 1}}} + {{2\xi} \over {\alpha + 1}} - \mu} \hfill \cr {{\eta _2} = \mu - {{2\xi} \over {\alpha + 1}}} \hfill \cr}
with ρ1 , ρ2 , Ψ1 , Ψ2 , γ1 , γ2 , δ1 , δ2 described as in Theorem 4 .
Proof
By using Lemma 3 and using properties of absolute value we have
| λ f ( m ) + μ f ( n ) 2 + ( 2 − λ − μ ) f ( ξ m + ( 1 − ξ ) n ) 2 − Γ ( α + 1 ) ( n − m ) α [ ( 1 − ξ ) 1 − α J [ ξ m + ( 1 − ξ ) n ] − α f ( m ) + ξ 1 − α J [ ξ m + ( 1 − ξ ) n ] + α f ( n ) ] | ≤ n − m 2 [ ( 1 − ξ ) ∫ 0 1 | 2 ( 1 − ξ ) t α − λ | | f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) m ) | dt + ξ ∫ 0 1 | μ − 2 ξ t α | | f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) n ) | dt ] .
\matrix{{\,\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\left({1 - \xi} \right)\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)} \right|dt} \right.} \hfill \cr {\left. {\,\,\, + \xi \int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)} \right|dt} \right].} \hfill \cr}
Using power mean inequality, it yields
| λ f ( m ) + μ f ( n ) 2 + ( 2 − λ − μ ) f ( ξ m + ( 1 − ξ ) n ) 2 − Γ ( α + 1 ) ( n − m ) α [ ( 1 − ξ ) 1 − α J [ ξ m + ( 1 − ξ ) n ] − α f ( m ) + ξ 1 − α J [ ξ m + ( 1 − ξ ) n ] + α f ( n ) ] | ≤ n − m 2 [ ( 1 − ξ ) ( ∫ 0 1 | 2 ( 1 − ξ ) t α − λ | dt ) 1 − 1 q × ( ∫ 0 1 | 2 ( 1 − ξ ) t α − λ | | f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) m ) | q dt ) 1 q + ξ ( ∫ 0 1 | μ − 2 ξ t α | dt ) 1 − 1 q ( ∫ 0 1 | μ − 2 ξ t α | | f ' ( t ( ξ m + ( 1 − ξ ) n ) + ( 1 − t ) n ) | q dt ) 1 q ] .
\matrix{{\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\matrix{{\left({1 - \xi} \right){{\left({\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt} \right)}^{1 - {1 \over q}}}} \cr {\times {{\left({\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|{{\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)m} \right)} \right|}^q}dt} \right)}^{{1 \over q}}}} \cr}} \right.} \hfill \cr {\left. {\,\,\, + \xi {{\left({\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|dt} \right)}^{1 - {1 \over q}}}{{\left({\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|{{\left| {{f^{'}}\left({t\left({\xi m + \left({1 - \xi} \right)n} \right) + \left({1 - t} \right)n} \right)} \right|}^q}dt} \right)}^{{1 \over q}}}} \right].} \hfill \cr}
By taking into account convexity of |f ′|q we get
| λ f ( m ) + μ f ( n ) 2 + ( 2 − λ − μ ) f ( ξ m + ( 1 − ξ ) n ) 2 − Γ ( α + 1 ) ( n − m ) α [ ( 1 − ξ ) 1 − α J [ ξ m + ( 1 − ξ ) n ] − α f ( m ) + ξ 1 − α J [ ξ m + ( 1 − ξ ) n ] + α f ( n ) ] | ≤ n − m 2 [ ( 1 − ξ ) ( ∫ 0 1 | 2 ( 1 − ξ ) t α − λ | dt ) 1 − 1 q × ( ∫ 0 1 | 2 ( 1 − ξ ) t α − λ | [ t | f ' ( ξ m + ( 1 − ξ ) n ) | q + ( 1 − t ) | f ' ( m ) | q ] dt ) 1 q + ξ ( ∫ 0 1 | μ − 2 ξ t α | dt ) 1 − 1 q × ( ∫ 0 1 | μ − 2 ξ t α | [ t | f ' ( ξ m + ( 1 − ξ ) n ) | q + ( 1 − t ) | f ' ( n ) | q ] dt ) 1 q ] .
\matrix{{\,\,\,\left| {{{\lambda f\left(m \right) + \mu f\left(n \right)} \over 2} + {{\left({2 - \lambda - \mu} \right)f\left({\xi m + \left({1 - \xi} \right)n} \right)} \over 2}} \right.} \hfill \cr {\left. {\,\,\, - {{\Gamma \left({\alpha + 1} \right)} \over {{{\left({n - m} \right)}^\alpha}}}\left[ {{{\left({1 - \xi} \right)}^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ -}}^\alpha f\left(m \right) + {\xi ^{1 - \alpha}}J_{{{\left[ {\xi m + \left({1 - \xi} \right)n} \right]}^ +}}^\alpha f\left(n \right)} \right]} \right|} \hfill \cr {\le {{n - m} \over 2}\left[ {\matrix{{\left({1 - \xi} \right){{\left({\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt} \right)}^{1 - {1 \over q}}}} \cr {\times {{\left({\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|\left[ {t{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + \left({1 - t} \right){{\left| {{f^{'}}\left(m \right)} \right|}^q}} \right]dt} \right)}^{{1 \over q}}}} \cr}} \right.} \hfill \cr {\left. {\matrix{{+ \xi {{\left({\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|dt} \right)}^{1 - {1 \over q}}}} \cr {\,\,\,\times {{\left({\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|\left[ {t{{\left| {{f^{'}}\left({\xi m + \left({1 - \xi} \right)n} \right)} \right|}^q} + \left({1 - t} \right){{\left| {{f^{'}}\left(n \right)} \right|}^q}} \right]dt} \right)}^{{1 \over q}}}} \cr}} \right].} \hfill \cr}
By making necessary computations we have
∫ 0 1 | 2 ( 1 − ξ ) t α − λ | dt = { ζ 1 , λ < 2 ( 1 − ξ ) ζ 2 , λ ≥ 2 ( 1 − ξ ) ∫ 0 1 | μ − 2 ξ t α | dt = { η 1 , μ < 2 ξ η 2 , μ ≥ 2 ξ .
\matrix{{\int_0^1 \left| {2\left({1 - \xi} \right){t^\alpha} - \lambda} \right|dt =} \hfill & {\left\{{\matrix{{{\zeta _1},} & {\lambda < 2\left({1 - \xi} \right)} \cr {{\zeta _2},} & {\lambda \ge 2\left({1 - \xi} \right)} \cr}} \right.} \hfill \cr {{\kern 30pt}\int_0^1 \left| {\mu - 2\xi {t^\alpha}} \right|dt =} \hfill & {\left\{{\matrix{{{\eta _1},} & {\mu < 2\xi} \cr {{\eta _2},} & {\mu \ge 2\xi} \cr}} \right..} \hfill \cr}