In [1, Definition 1.9], the concept “s-geometrically convex function” was introduced.
Making use of [1, Lemma 2.1], Hölder’s integral inequality, and other analytic techniques, some inequalities of Hermite-Hadamard type were established. However, there are some vital errors appeared in main results of the paper [1].
The aim of this paper is to correct these errors and we now start off to correct them.
Correction to Theorem3.1. Let f:I⊂ℝ+→ℝ be a differentiable function on I∘ such that f′∈L([a,b]) for 0<a<b<∞. If |f′(x)|q is s-geometrically convex and monotonically decreasing on [a,b] for q≥1 and s∈(0,1], then
(1)|f(a+b2)-1b-a∫abf(x)dx|≤b-a4(12)1-1/qG1(s,q;g1(α),g2(α)),(2)|f(a)+f(b)2-1b-a∫abf(x)dx|≤b-a4(12)1-1/qG1(s,q;g2(α),g1(α)),
where
(3)g1(α)={12,α=1,αlnα-α+1ln2α,α≠1,g2(α)={12,α=1,α-lnα-1ln2α,α≠1,(4)α=|f′(b)f′(a)|sq/2,(5)G1(s,q;g1(α),g2(α))={|f′(a)|s[g1(α)]1/q+|f′(a)f′(b)|s/2×[g2(α)]1/q,|f′(a)|≤1,|f′(a)|[g1(α)]1/q+|f′(a)|1-s/2×|f′(b)|s/2[g2(α)]1/q,|f′(b)|≤1≤|f′(a)|,|f′(a)||f′(b)|1-s[g1(α)]1/q+|f′(a)f′(b)|1-s/2×[g2(α)]1/q,1≤|f′(b)|.
Proof.
Since |f′|q is s-geometrically convex and monotonically decreasing on [a,b], using Lemma 2.1 and Hölder’s inequality gives
(6)|f(a+b2)-1b-a∫abf(x)dx|≤b-a4∫01[t|f′((1-t)a+ta+b2)|+(1-t)|f′((1-t)a+b2+tb)|]dt≤b-a4{(∫01tdt)1-1/q×[∫01t|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt]1/q+(∫01(1-t)dt)1-1/q×[∫01(1-t)|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt]1/q}=b-a4(12)1-1/q{∫01[∫01t|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt]1/q+[∫01(1-t)|f′(a)|q((1-t)/2)s×|f′(b)|q((1+t)/2)sdt∫01]1/q}.
Let 0<μ≤1≤η and 0<s,t≤1. Then
(7)μts≤μst,ηts≤ηst+1-s.
When |f′(a)|≤1, we have
(8)∫01t|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt≤∫01t|f′(a)|sq((2-t)/2)|f′(b)|sq(t/2)dt=|f′(a)|sqg1(α),(9)∫01(1-t)|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt≤∫01(1-t)|f′(a)|sq((1-t)/2)|f′(b)|sq((1+t)/2)dt=|f′(a)f′(b)|sq/2g2(α).
When |f′(b)|≤1≤|f′(a)|, by (7), we obtain
(10)∫01t|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt≤∫01t|f′(a)|q[s(2-t)/2+1-s]|f′(b)|sqt/2dt=|f′(a)|qg1(α),(11)∫01(1-t)|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt≤∫01(1-t)|f′(a)|q[s(1-t)/2+1-s]|f′(b)|sq(1+t)/2dt=|f′(a)|(1-s/2)q|f′(b)|sq/2g2(α).
When 1≤|f′(b)|, by (7), we have
(12)∫01t|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt≤|f′(a)|q|f′(b)|(1-s)qg1(α),(13)∫01(1-t)|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt≤|f′(a)f′(b)|(1-s/2)qg2(α).
Substituting (8) to (13) into (6) yields inequality (1).
Since |f′|q is s-geometrically convex and monotonically decreasing on [a,b], by Lemma 2.1 and Hölder’s inequality, we obtain
(14)|f(a)+f(b)2-1b-a∫abf(x)dx|≤b-a4∫01[(1-t)|f′((1-t)a+ta+b2)|+t|f′((1-t)a+b2+tb)|]dt≤b-a4(12)1-1/q{|f′(b)|q((1+t)/2)sdt∫01]1/q∫01[∫01(1-t)|f′(a)|q((2-t)/2)s×|f′(b)|q(t/2)sdt∫01]1/q+[∫01t|f′(a)|q((1-t)/2)s×|f′(b)|q((1+t)/2)sdt∫01]1/q∫01}.
When |f′(a)|≤1, we have
(15)∫01(1-t)|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt≤|f′(a)|sqg2(α),(16)∫01t|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt≤|f′(a)f′(b)|sq/2g1(α).
When |f′(b)|≤1≤|f′(a)|, by (7), we obtain
(17)∫01(1-t)|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt≤|f′(a)|qg2(α),(18)∫01t|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt≤|f′(a)|(1-s/2)q|f′(b)|sq/2g1(α).
When 1≤|f′(b)|, by (7), we have
(19)∫01(1-t)|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt≤|f′(a)|q|f′(b)|(1-s)qg2(α),(20)∫01t|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt≤|f′(a)f′(b)|(1-s/2)qg1(α).
Substituting (15) to (20) into (14) leads to inequality (2). Theorem 3.1 is thus proved.
Correction to Corollary 3.2. Under the conditions of Theorem 3.1,
when q=1,
(21)|f(a+b2)-1b-a∫abf(x)dx|≤b-a4G1(s,1;g1(α),g2(α)),|f(a)+f(b)2-1b-a∫abf(x)dx|≤b-a4G1(s,1;g2(α),g1(α));
when s=1,
(22)|f(a+b2)-1b-a∫abf(x)dx|≤b-a4(12)1-1/qG1(1,q;g1(α),g2(α)),|f(a)+f(b)2-1b-a∫abf(x)dx|≤b-a4(12)1-1/qG1(1,q;g2(α),g1(α)).
Correction to Theorem3.3. Let f:I⊂ℝ+→ℝ be a differentiable function on I∘ such that f′∈L([a,b]) for 0<a<b<∞. If |f′(x)|q is s-geometrically convex and monotonically decreasing on [a,b] for q>1 and s∈(0,1], then
(23)|f(a+b2)-1b-a∫abf(x)dx|≤b-a4(q-12q-1)1-1/qG2(s,q;g3(α)),(24)|f(a)+f(b)2-1b-a∫abf(x)dx|≤b-a4(q-12q-1)1-1/qG2(s,q;g3(α)),
where α is the same as in (4),
(25)g3(α)={1,α=1,α-1lnα,α≠1,G2(s,q;g3(α))={[|f′(a)|s+|f′(a)f′(b)|s/2][g3(α)]1/q,|f′(a)|≤1,[|f′(a)|+|f′(a)|1-s/2|f′(b)|s/2]×[g3(α)]1/q,|f′(b)|≤1≤|f′(a)|,[|f′(a)||f′(b)|1-s+|f′(a)f′(b)|1-s/2]×[g3(α)]1/q,1≤|f′(b)|.
Proof.
Since |f′|q is s-geometrically convex and monotonically decreasing on [a,b], by Lemma 2.1 and Hölder’s inequality, we have
(26)|f(a+b2)-1b-a∫abf(x)dx|≤b-a4(q-12q-1)1-1/q×{[∫01|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt]1/q+[∫01|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt]1/q},(27)|f(a)+f(b)2-1b-a∫abf(x)dx|≤b-a4(q-12q-1)1-1/q×{[∫01|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt]1/q+[∫01|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt]1/q}.
When |f′(a)|≤1, we have
(28)∫01|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt≤|f′(a)|sqg3(α),(29)∫01|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt≤|f′(a)f′(b)|sq/2g3(α).
When |f′(b)|≤1≤|f′(a)|, by (7), we obtain
(30)∫01|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt≤|f′(a)|qg3(α),(31)∫01|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt≤|f′(a)|(1-s/2)q|f′(b)|sq/2g3(α).
When 1≤|f′(b)|, by (7), we have
(32)∫01|f′(a)|q((2-t)/2)s|f′(b)|q(t/2)sdt≤|f′(a)|q|f′(b)|(1-s)qg3(α),(33)∫01|f′(a)|q((1-t)/2)s|f′(b)|q((1+t)/2)sdt≤|f′(a)f′(b)|(1-s/2)qg3(α).
Substituting (28) to (33) into (26) and (27) results in inequalities (23) and (24). Theorem 3.3 is thus proved.
Correction to Corollary3.4. Under the conditions of Theorem 3.3, when s=1, we have
(34)|f(a+b2)-1b-a∫abf(x)dx|≤b-a4(q-12q-1)1-1/qG2(1,q;g3(α)),(35)|f(a)+f(b)2-1b-a∫abf(x)dx|≤b-a4(q-12q-1)1-1/qG2(1,q;g3(α)).
Correction to Theorem4.1. Let 0<a<b≤1, 0<s<1, and q≥1. Then
(36)0<[A(a,b)]s-[Ls(a,b)]s≤(b-a)1-1/qs8[4s(1-s)qL(a,b)]1/q×{[bs(1-s)q/2-L(as(1-s)q/2,bs(1-s)q/2)]1/qas-1b(s-1)(1-s/2)×[L(as(1-s)q/2,bs(1-s)q/2)-as(1-s)q/2]1/q+a(s-1)(1-s/2)bs-1×[bs(1-s)q/2-L(as(1-s)q/2,bs(1-s)q/2)]1/q},(37)0<[Ls(a,b)]s-A(as,bs)≤(b-a)1-1/qs8[4s(1-s)qL(a,b)]1/q×{[L(as(1-s)q/2,bs(1-s)q/2)-as(1-s)q/2]1/qas-1b(s-1)(1-s/2)×[bs(1-s)q/2-L(as(1-s)q/2,bs(1-s)q/2)]1/q+a(s-1)(1-s/2)bs-1×[L(as(1-s)q/2,bs(1-s)q/2)-as(1-s)q/2]1/q},
where
(38)A(a,b)=a+b2,L(a,b)=b-alnb-lna,(39)Lp(a,b)=[bp+1-ap+1(p+1)(b-a)]1/p
for a≠b and p∈ℝ with p≠0,-1 are the arithmetic, logarithmic, and generalized logarithmic means, respectively.
If q=1, then
(40)0<[A(a,b)]s-[Ls(a,b)]s≤L(a,b)2(1-s)×{[bs(1-s)/2-L(as(1-s)/2,bs(1-s)/2)]as-1b(s-1)(1-s/2)×[L(as(1-s)/2,bs(1-s)/2)-as(1-s)/2]+a(s-1)(1-s/2)bs-1×[bs(1-s)2-L(as(1-s)2,bs(1-s)2)]},(41)0<[Ls(a,b)]s-A(as,bs)≤L(a,b)2(1-s){as-1b(s-1)(1-s/2)×[bs(1-s)/2-L(as(1-s)/2,bs(1-s)/2)]+a(s-1)(1-s/2)bs-1×[L(as(1-s)/2,bs(1-s)/2)-as(1-s)/2]}.
Proof.
Let 0<s<1, q≥1, and f(x)=xs/s for x∈(0,1]. Then the function |f′(x)|q is s-geometrically convex on (0,1] for 0<s<1,
(42)|f′(a)|=as-1>bs-1=|f′(b)|≥1,
and α=(a/b)s(1-s)q/2. Therefore,
(43)g1(α)=2s(1-s)qbs(1-s)q/2(lnb-lna)×[L(as(1-s)q/2,bs(1-s)q/2)-as(1-s)q/2],(44)g2(α)=2s(1-s)qbs(1-s)q/2(lnb-lna)×[bs(1-s)q/2-L(as(1-s)q/2,bs(1-s)q/2)].
By Theorem 3.1, Theorem 4.1 is thus proved.
Correction to Theorem4.2. Let 0<a<b≤1, 0<s<1, and q>1. Then
(45)0<[A(a,b)]s-[Ls(a,b)]s≤(b-a)s4(q-12q-1)1/q×[as-1b(s-1)(1-s/2)+a(s-1)(1-s/2)bs-1]×[L(as(1-s)q/2,bs(1-s)q/2)]1/q,(46)0<[Ls(a,b)]s-A(as,bs)≤(b-a)s4(q-12q-1)1/q×[as-1b(s-1)(1-s/2)+a(s-1)(1-s/2)bs-1]×[L(as(1-s)q/2,bs(1-s)q/2)]1/q.
Proof.
It is easy to see that
(47)g3(α)=1qbs(1-s)q/2L(as(1-s)q/2,bs(1-s)q/2).
Hence, by Theorem 3.3, Theorem 4.2 is thus proved.
Remark.
By the way, all the powers 1-(3/q) which appeared four times in [2, Theorem 4.2 and Corollary 4.2] should be corrected as 3(1-(1/q)), respectively.
Acknowledgments
The authors would like to thank Professor Feng Qi in China for his valuable contributions to these corrections. This work was supported in part by the NNSF of China under Grant no. 11361038 and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant no. NJZY14191 and NJZY13159, China.
ZhangT.-Y.JiA.-P.QiF.On integral inequalities of Hermite-Hadamard type for s-geometrically convex functions201220121456058610.1155/2012/560586XiB.-Y.BaiR.-F.QiF.Hermite-Hadamard type inequalities for the m- and (α,m)-geometrically convex functions201284326126910.1007/s00010-011-0114-xMR2996418