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Means of different types play significant role in different fields of sciences through their applications. For instance it has been observed harmonic means have applications in electrical circuits theory. To be more precise, the total resistance of a set of parallel resistors is just half of harmonic means of the total resistors, for details, see [3]. Recently many researchers have extensively utilized different types of means in theory of convexity. Consequently a number of new and novel extensions of classical convexity have been proposed in the literature. For some recent studies, see [4,5,21,22]. We now recall some preliminary concepts and results pertaining to convexity and for its other extensions.
Definition 1.1 ([18]). (AA-convex functions) A function X:C⊆R→R is said to be AA-convex, if
(1−μ)X(x)+μX(y)≥X((1−μ)x+ty),∀x,y∈C,μ∈[0,1], |
where C is a convex set.
Definition 1.2 ([18]). (GG-convex functions) A function X:G⊆R+→R+ is said to be GG-convex, if
X1−μ(x)Xμ(y)≥X(x1−μyμ),∀x,y∈G,μ∈[0,1], |
where G is a geometric convex set.
Definition 1.3 ([13]). (HH-convex functions) A function X:H⊆R+→R is said to be HH-convex, if
X(x)X(y)μX(x)+(1−μ)X(y)≥X(xy(1−μ)x+ty),∀x,y∈H,μ∈[0,1], |
where H is a harmonic convex set.
For some other useful details, see [18]. Convexity theory also played significant role in the development of theory of inequalities. Many known results are obtained directly using the functions having convexity property. Hermite and Hadamard presented independently a result which now a days known as Hermite-Hadamard's inequality. This result is very simple in nature but very powerful, as it provides us a necessary and sufficient condition for a function to be convex. It reads as: Let X:I⊆R→R be a convex function, then
X(c+d2)≤1d−cd∫cX(x)dx≤X(c)+X(d)2. |
Dragomir et al. [8] written a very interesting detailed monograph on Hermite-Hadamard's inequality and its applications. Interested readers may find useful details in it. In recent years several famously known researchers from all over the world have studied the result of Hermite and Hadamard intensively. For more details, see [4,6,7,9,10,17,20]. This result has also been generalized for other classes of convex functions, for instance, see [8,11,12,14,18,22].
Fractional calculus [15,16] has played an important role in various scientific fields since it is a good tool to describe long-memory processes. Sarikaya et al. [24] used the concepts of fractional calculus and obtained new refinements of fractional Hermite-Hadamard like inequalities. This article of Sarikaya et al. opened a new venue of research. Consequently several new generalizations of Hermite-Hadamard's inequality have been obtained using the fractional calculus concepts.
Recently many authors have shown their special interest in utilizing the concepts of quantum calculus for obtaining q-analogues of different integral inequalities. For some basic definitions and recent studies, see [1,2,19,23,25,26]. The main objective of this article is to introduce the notion of M-convex functions. This class can be viewed as novel extension of the classical definition of convexity. We link this class with Hermite-Hadamard's inequality and obtain several new variants of this famous result. We also obtain the fractional and quantum analogues of the obtained results. We expect that the results of this paper may stimulate further research in this direction.
In this section, we introduce the notions of M-convex functions, log-M-convex and quasi M-convex functions. First of all for the sake of simplicity, we take G=√cd and A=c+d2.
Definition 2.1. A function X:D⊆R+→R+ is said to be M-convex function, if
X((1−μ)G+μA)≤(1−μ)X(G)+tX(A),∀c,d∈D,μ∈[0,1]. |
Definition 2.2. A function X:D⊆R+→R+ is said to be log-M-convex function, if
X((1−μ)G+μA)≤X1−μ(G)Xμ(A),∀c,d∈D,μ∈[0,1]. |
Definition 2.3. A function X:D⊆R+→R+ is said to be quasi M-convex function, if
X((1−μ)G+μA)≤max{X(G),X(A)},∀c,d∈D,μ∈[0,1]. |
We now derive a new auxiliary result which play a key role in the development of our coming results.
Lemma 3.1. Let X:I∘⊆R+→R+ be a differentiable function on I∘, c,d∈I∘ with c<d. If X′∈L[c,d], then
X(G)+X(A)2−2(√d−√c)2A∫GX(x)dx=(√d−√c)241∫0(1−2μ)X′(μG+(1−μ)A)dμ. |
Proof. It suffices to show that
1∫0(1−2μ)X′(μG+(1−μ)A)dμ=2X(G)+X(A)(√d−√c)2−8(√d−√c)4A∫GX(x)dx. |
This implies
(√d−√c)241∫0(1−2μ)X′(μG+(1−μ)A)dμ=X(G)+X(A)2−2(√d−√c)2A∫GX(x)dx. |
This completes the proof.
Now utilizing Lemma 3.1, we derive our next results.
Theorem 3.2. Let X:I∘⊆R+→R+ be a differentiable function on I∘, c,d∈I∘ with c<d and X′∈L[c,d]. If |X′| is M-convex function, then
|X(G)+X(A)2−2(√d−√c)2A∫GX(x)dx|≤(√d−√c)216[|X′(G)|+|X′(A)|]. |
Proof. Using Lemma 3.1, property of the modulus and the fact that |X′| is M-convex function, we have
|X(G)+X(A)2−2(√d−√c)2A∫GX(x)dx|≤(√d−√c)241∫0|1−2μ||X′(μG+(1−μ)A)|dμ≤(√d−√c)241∫0|1−2μ|[μ|X′(G)|+(1−μ)|X′(A)|]dμ=(√d−√c)216[|X′(G)|+|X′(A)|]. |
This completes the proof.
If we apply Theorem 3.2 for log-M-convex functions, then
Theorem 3.3. Let X:I∘⊆R+→R+ be a differentiable function on I∘, c,d∈I∘ with c<d and X′∈L[c,d], If |X′| is decreasing and log-M-convex function, then
|X(G)+X(A)2−2(√d−√c)2A∫GX(x)dx|≤(√d−√c)24[−2+4√w−2w−logw+wlogwlogw2], |
where w=|X′(G)||X′(A)|.
Theorem 3.4. Let X:I∘⊆R+→R+ be a differentiable function on I∘, c,d∈I∘ with c<d and X′∈L[c,d]. If |X′|q, where 1p+1q=1 is M-convex function, then
|X(G)+X(A)2−2(√d−√c)2A∫GX(x)dx|≤(√d−√c)24(1p+1)1p(|X′(G)|q+|X′(A)|q2)1q. |
Proof. Using Lemma 3.1, Holder's inequality and the fact that |X′|q is M-convex functions, we have
|X(G)+X(A)2−2(√d−√c)2A∫GX(x)dx|≤(√d−√c)24(1∫0|1−2μ|pdμ)1p(1∫0|X′(μG+(1−μ)A)|dμ)1q≤(√d−√c)24(1p+1)1p(1∫0[μ|X′(G)|q+(1−μ)|X′(A)|q]dμ)1q=(√d−√c)24(1p+1)1p(|X′(G)|q+|X′(A)|q2)1q. |
This completes the proof.
Theorem 3.5. Let X:I∘⊆R+→R+ be a differentiable function on I∘, c,d∈I∘ with c<d and X′∈L[c,d]. If |X′|q, where q≥1 is M-convex function, then
|X(G)+X(A)2−2(√d−√c)2A∫GX(x)dx|≤(√d−√c)28(|X′(G)|q+|X′(A)|q2)1q. |
Proof. Using Lemma 3.1, power mean inequality and the fact that |X′| is M-convex functions, we have
|X(G)+X(A)2−2(√d−√c)2A∫GX(x)dx|≤(√d−√c)24(1∫0|1−2μ|dμ)1−1q(1∫0|1−2μ||X′(μG+(1−μ)A)|dμ)1q≤(√d−√c)24(12)1−1q(1∫0|1−2μ|[μ|X′(G)|q+(1−μ)|X′(A)|q]dμ)1q=(√d−√c)28(|X′(G)|q+|X′(A)|q2)1q. |
This completes the proof.
In this section, we derive some fractional estimates of Hermite-Hadamard like inequalities using M-convex functions. Before that we recall basic definition of Riemann-Liouville fractional integrals.
Definition 4.1 ([15]). Let X∈L[c,d], where c≥0. The Riemann-Liouville integrals Jνc+X and Jνd−X, of order ν>0, are defined by
Jνc+X(x)=1Γ(ν)∫xc(x−μ)ν−1X(μ)dμ, for x>c |
and
Jνd−X(x)=1Γ(ν)∫dx(μ−x)ν−1X(μ)dμ, for x<d, |
respectively. Here, Γ(ν)=∫∞0e−μμν−1dμ is the Gamma function. We also make the convention
J0c+X(x)=J0d−X(x)=X(x). |
We now derive a new auxiliary result utilizing the definition of Riemann-Liouville fractional integrals.
Lemma 4.1. Let X:I∘⊆R+→R+ be a differentiable function. If X′∈L[c,d], then
X(G)+X(A)2−2α−1Γ(α+1)(√d−√c)2[Jα(A)−X(G)+Jα(G)+X(A)]=(√d−√c)241∫0[(1−μ)α−μα]X′(μG+(1−μ)A)dμ. |
Proof. It suffices to show that
I=1∫0[(1−μ)α−μα]X′(μG+(1−μ)A)dμ=1∫0(1−μ)αX′(μG+(1−μ)A)dμ−1∫0μαX′(μG+(1−μ)A)dμ=I1−I2. | (4.1) |
Now using change of variable technique and definition of Riemann-Liouville fractional integrals, we have
I1=1∫0(1−μ)αX′(μG+(1−μ)A)dμ=2(√d−√c)2X(A)−2α+1Γ(α+1)(√d−√c)2(α+1)1Γ(α)A∫G(x−G)α−1X(x)dx=2(√d−√c)2X(A)−2α+1Γ(α+1)(√d−√c)2(α+1)Jα(A)−X(G). | (4.2) |
Similarly
I2=1∫0μαX′(μG+(1−μ)A)dμ=−2(√d−√c)2X(G)+2α+1Γ(α+1)(√d−√c)2(α+1)Jα(G)+X(A). | (4.3) |
Combining (4.1), (4.2) and (4.3) completes the proof. Now using Lemma 4.1, we derive our next results.
Theorem 4.2. Let X:I∘⊆R+→R+ be a differentiable function and X′∈L[c,d]. If |X′| is M-convex function, then
|X(G)+X(A)2−2α−1Γ(α+1)(√d−√c)2[Jα(A)−X(G)+Jα(G)+X(A)]|≤(√d−√c)24(α+1)(1−12α)[|X′(a)|+|X′(b)|]. |
Proof. Using Lemma 4.1 and the property of modulus, we have
|X(G)+X(A)2−2α−1Γ(α+1)(√d−√c)2[Jα(A)−X(G)+Jα(G)+X(A)]|≤1∫0(√d−√c)24|(1−μ)α−μα||X′(μG+(1−μ)A)|dμ. |
Since it is given that |X′| is M-convex function, so we have
|X(G)+X(A)2−2α−1Γ(α+1)(√d−√c)2[Jα(A)−X(G)+Jα(G)+X(A)]|≤1∫0(√d−√c)24|(1−μ)α−μα|[μ|X′(G)|+(1−μ)|X′(A)|]dμ=(√d−√c)24[|X′(G)|1∫0μ|(1−μ)α−μα|dμ+|X′(A)|1∫0(1−μ)|(1−μ)α−μα|dμ]=(√d−√c)24(α+1)(1−12α)[|X′(a)|+|X′(b)|]. |
This completes the proof.
In this section, we derive some quantum analogues of Hermite-Hadamard like inequalities using M-convex functions. Before proceeding, let us recall some basics of quantum calculus. Tariboon et al. [25] defined the q-integral as follows:
Definition 5.1 ([25]). Let X:I⊂R→R be a continuous function. Then q-integral on I is defined as:
∫xaX(μ)adqμ=(1−q)(x−a)∞∑n=0qnX(qnx+(1−qn)a), | (5.1) |
for x∈J.
The following result will play significant role in main results of the section.
Lemma 5.1 ([25]). Let α∈R∖{−1}, then
x∫a(μ−a)αadqμ=(1−q1−qα+1)(x−a)α+1. |
Lemma 5.2. Let X:I∘⊆R+→R+ be a q-differentiable function on I∘, c,d∈I∘ with c<d. If DqX is an integrable function with 0<q<1, then
2(√d−√c)2A∫GX(μ)dqμ−qf(G)+X(A)1+q=q(√d−√c)22(1+q)1∫0(1−(1+q)μ)Dq((1−μ)G+μA)dqμ. |
Proof. It suffices to show that
1∫0(1−(1+q)μ)Dq((1−μ)G+μA)dqμ=2(√d−√c)21∫0(X((1−μ)G+μA)−X((1−qμ)G+qμA)(1−q)μ)dqμ−2(1+q)(√d−√c)21∫0μ(X((1−μ)G+μA)−X((1−qμ)G+qμA)(1−q)μ)dqμ=2(√d−√c)2[∞∑n=0X((1−qn)G+qnA)−∞∑n=0X((1−qn+1)G+qn+1A)]−2(1+q)(√d−√c)2[∞∑n=0qnX((1−qn)G+qnA)−∞∑n=0qnX((1−qn+1)G+qn+1A)]=2(√d−√c)2[X(A)−X(G)]−2(1+q)(√d−√c)2∞∑n=0qnX((1−qn)G+qnA)+2(1+q)q(√d−√c)2∞∑n=1qnX((1−qn)G+qnA)=2(√d−√c)2[X(A)−X(G)]−2(1+q)(√d−√c)2∞∑n=0qnX((1−qn)G+qnA)+2(1+q)q(√d−√c)2[X(A)−X(A)+∞∑n=1qnX((1−qn)G+qnA)]=−2q(√d−√c)2[qf(G)+X(A)]+4(1+q)q(√d−√c)4A∫GX(μ)dqμ. |
This completes the proof.
Now using Lemma 5.2, we derive our next results.
Theorem 5.3. Let X:I∘⊆R+→R+ be a q-differentiable function on I∘, c,d∈I∘ with c<d and DqX is an integrable function with 0<q<1. If |DqX| is M-convex, then
|2(√d−√c)2A∫GX(μ)dqμ−qf(G)+X(A)1+q|≤q(√d−√c)22(1+q)4(1+q+q2){(1+3q2+2q3)|DqX(G)|+(1+4q+q2)|DqX(A)|}. |
Proof. Using Lemma 5.2 and the given hypothesis of the theorem, we have
|2(√d−√c)2A∫GX(μ)dqμ−qf(G)+X(A)1+q|=|q(√d−√c)22(1+q)1∫0(1−(1+q)μ)DqX((1−μ)G+μA)dqμ|≤q(√d−√c)22(1+q)1∫0|1−(1+q)μ|[(1−μ)|DqX(G)|+μDq|DqX(A)|]dqμ=q(√d−√c)22(1+q){|DqX(G)|1∫0(1−μ)|1−(1+q)μ|dqμ+|DqX(A)|1∫0μ|1−(1+q)μ|dqμ}=q(√d−√c)22(1+q)4(1+q+q2){(1+3q2+2q3)|DqX(G)|+(1+4q+q2)|DqX(A)|}. |
This completes the proof.
Theorem 5.4. Let X:I∘⊆R+→R+ be a q-differentiable function on I∘, c,d∈I∘ with c<d and DqX is an integrable function with 0<q<1. If |DqX|r is M-convex, where r>1, then
|2(√d−√c)2A∫GX(μ)dqμ−qf(G)+X(A)1+q|≤q(√d−√c)22(1+q)(2q(1+q)2)1−1r(q(1+3q2+2q3)(1+q)3(1+q+q2)|DqX(G)|r+q(1+4q+q2)(1+q)3(1+q+q2)|DqX(A)|r)1r. |
Proof. Using Lemma 5.2, power-mean inequality and the given hypothesis of the theorem, we have
|2(√d−√c)2A∫GX(μ)dqμ−qf(G)+X(A)1+q|=|q(√d−√c)22(1+q)1∫0(1−(1+q)μ)DqX((1−μ)G+μA)dqμ|≤q(√d−√c)22(1+q)(1∫0|1−(1+q)μ|dqμ)1−1r×(1∫0|1−(1+q)μ|[(1−μ)|DqX(G)|r+μ|DqX(A)|r]dqμ)1r=q(√d−√c)22(1+q)(2q(1+q)2)1−1r(q(1+3q2+2q3)(1+q)3(1+q+q2)|DqX(G)|r+q(1+4q+q2)(1+q)3(1+q+q2)|DqX(A)|r)1r. |
This completes the proof.
In this article, we have introduced the notions of M-convex functions, log-M-convex functions and quasi M-convex functions. We have discussed these classes in context with integral inequalities of Hermite-Hadamard type. We have also obtained some new fractional and quantum versions of these results. It is worth to mention here that essentially using the techniques of this article one can easily obtain extensions of Iynger type inequalities using the class of quasi M-convex functions. We hope that the ideas and techniques of this paper will inspire interested readers working in the field.
Authors are thankful to the editor and anonymous referees for their valuable comments and suggestions. First and second authors are thankful for the support of HEC project (No. 8081/Punjab/NRPU/R&D/HEC/2017).
The authors declare no conflicts of interest.
[1] |
Arbogast JW, Darmanyan AP, Foote CS, et al. (1991) Photophysical properties of sixty atom carbon molecule (C60). J Phys Chem 95: 11–12. doi: 10.1021/j100154a006
![]() |
[2] |
Ching WY, Huang MZ, Xu YN, et al. (1991) First-principles calculation of optical properties of the carbon sixty-atom molecule in the fcc. lattice. Phys Rev Lett 67: 2045–2048. doi: 10.1103/PhysRevLett.67.2045
![]() |
[3] |
Maser W, Roth S, Anders J, et al. (1992) P-Type doping of C60 fullerene films. Synth Met 51: 103–108. doi: 10.1016/0379-6779(92)90259-L
![]() |
[4] | Sun Y-P, Lawson GE, Riggs JE, et al. (1998) Photophysical and Nonlinear Optical Properties of [60]Fullerene Derivatives. J Phys Chem A 102: 5520–5528. |
[5] |
Accorsi G, Armaroli N (2010) Taking Advantage of the Electronic Excited States of [60]-Fullerenes. J Phys Chem C 114: 1385–1403. doi: 10.1021/jp9092699
![]() |
[6] | Allemand PM, Khemani KC, Koch A, et al. (1991) Organic molecular soft ferromagnetism in a fullerene C60. Science 253: 301–303. |
[7] | Stephens PW, Cox D, Lauher JW, et al. (1992) Lattice structure of the fullerene ferromagnet TDAE-C60. Nature 355: 331–332. |
[8] | Hebard AF, Rosseinsky MJ, Haddon RC, et al. (1991) Superconductivity at 18 K in potassium-doped fullerene (C60). Nature 350: 600–601. |
[9] | Dubois D, Moninot G, Kutner W, et al. (1992) Electroreduction of Buckminsterfullerene, C60, in aprotic solvents. Solvent, supporting electrolyte, and temperature effects. J Phys Chem 96: 7137–7145. |
[10] |
Schon TB, Di Carmine PM, Seferos DS (2014) Polyfullerene Electrodes for High Power Supercapacitors. Adv Energy Mater 4: 1301509–1301515. doi: 10.1002/aenm.201301509
![]() |
[11] |
Pupysheva OV, Farajian AA, Yakobson BI (2008) Fullerene Nanocage Capacity for Hydrogen Storage. Nano Lett 8: 767–774. doi: 10.1021/nl071436g
![]() |
[12] |
Nadtochenko VA, Vasil'ev IV, Denisov NN, et al. (1993) Photophysical properties of fullerene C60: picosecond study of intersystem crossing. J Photochem Photobiol, A 70: 153–156. doi: 10.1016/1010-6030(93)85035-7
![]() |
[13] | Foote CS (1994) Photophysical and photochemical properties of fullerenes. Electron Transfer I. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 347–363. |
[14] | Sun R, Jin C, Zhang X, et al. (1994) Photophysical properties of C60. Wuli 23: 83–87. |
[15] |
Qu B, Chen SM, Dai LM (2000) Simulation analysis of ESR spectrum of polymer alkyl-C60 radicals formed by photoinitiated reactions of low-density polyethylene. Appl Magn Reson 19: 59–67. doi: 10.1007/BF03162261
![]() |
[16] |
Guldi DM, Asmus K-D (1997) Photophysical Properties of Mono- and Multiply-Functionalized Fullerene Derivatives. J Phys Chem A 101: 1472–1481. doi: 10.1021/jp9633557
![]() |
[17] |
McEwen CN, McKay RG, Larsen BS (1992) C60 as a radical sponge. J Am Chem Soc 114: 4412–4414. doi: 10.1021/ja00037a064
![]() |
[18] | Tzirakis MD, Orfanopoulos M (2013) Radical Reactions of Fullerenes: From Synthetic Organic Chemistry to Materials Science and Biology. Chem Rev 113: 5262–5321. |
[19] | Krusic PJ, Wasserman E, Keizer PN, et al. (1991) Radical reactions of C60. Science 254: 1183–1185. |
[20] |
Krusic PJ, Wasserman E, Parkinson BA, et al. (1991) Electron spin resonance study of the radical reactivity of C60. J Am Chem Soc 113: 6274–6275. doi: 10.1021/ja00016a056
![]() |
[21] |
Wu S-H, Sun W-Q, Zhang D-W, et al. (1998) Reaction of [60]fullerene with trialkylphosphine oxide. Tetrahedron Lett 39: 9233–9236. doi: 10.1016/S0040-4039(98)02131-5
![]() |
[22] |
Cheng F, Yang X, Fan C, et al. (2001) Organophosphorus chemistry of fullerene: synthesis and biological effects of organophosphorus compounds of C60. Tetrahedron 57: 7331–7335. doi: 10.1016/S0040-4020(01)00670-6
![]() |
[23] |
Cheng F, Yang X, Zhu H, et al. (2000) Synthesis and optical properties of tetraethyl methano[60]fullerenediphosphonate. Tetrahedron Lett 41: 3947–3950. doi: 10.1016/S0040-4039(00)00491-3
![]() |
[24] |
Liu Z-B, Tian J-G, Zang W-P, et al. (2003) Large optical nonlinearities of new organophosphorus fullerene derivatives. Appl Opt 42: 7072–7076. doi: 10.1364/AO.42.007072
![]() |
[25] |
Ford WT, Nishioka T, Qiu F, et al. (1999) Structure Determination and Electrochemistry of Products from the Radical Reaction of C60 with Azo(bisisobutyronitrile). J Org Chem 64: 6257–6262. doi: 10.1021/jo990346w
![]() |
[26] |
Ford WT, Nishioka T, Qiu F, et al. (2000) Dimethyl Azo(bisisobutyrate) and C60 Produce 1,4- and 1,16-Di(2-carbomethoxy-2-propyl)-1,x-dihydro[60]fullerenes. J Org Chem 65: 5780–5784. doi: 10.1021/jo000686d
![]() |
[27] |
Shustova NB, Peryshkov DV, Kuvychko IV, et al. (2011) Poly(perfluoroalkylation) of Metallic Nitride Fullerenes Reveals Addition-Pattern Guidelines: Synthesis and Characterization of a Family of Sc3N@C80(CF3)n (n = 2-16) and Their Radical Anions. J Am Chem Soc 133: 2672–2690. doi: 10.1021/ja109462j
![]() |
[28] |
Shu C, Slebodnick C, Xu L, et al. (2008) Highly Regioselective Derivatization of Trimetallic Nitride Templated Endohedral Metallofullerenes via a Facile Photochemical Reaction. J Am Chem Soc 130: 17755–17760. doi: 10.1021/ja804909t
![]() |
[29] |
Shu C, Cai T, Xu L, et al. (2007) Manganese(III)-Catalyzed Free Radical Reactions on Trimetallic Nitride Endohedral Metallofullerenes. J Am Chem Soc 129: 15710–15717. doi: 10.1021/ja0768439
![]() |
[30] |
Shustova NB, Popov AA, Mackey MA, et al. (2007) Radical Trifluoromethylation of Sc3N@C80. J Am Chem Soc 129: 11676–11677. doi: 10.1021/ja074332g
![]() |
[31] |
Cardona CM, Kitaygorodskiy A, Echegoyen L (2005) Trimetallic nitride endohedral metallofullerenes: Reactivity dictated by the encapsulated metal cluster. J Am Chem Soc 127: 10448–10453. doi: 10.1021/ja052153y
![]() |
[32] | Yu G, Gao J, Hummelen JC, et al. (1995) Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions. Science 270: 1789–1791. |
[33] |
Troshin PA, Hoppe H, Renz J, et al. (2009) Material Solubility-Photovoltaic Performance Relationship in the Design of Novel Fullerene Derivatives for Bulk Heterojunction Solar Cells. Adv Funct Mater 19: 779–788. doi: 10.1002/adfm.200801189
![]() |
[34] |
Jiao F, Liu Y, Qu Y, et al. (2010) Studies on anti-tumor and antimetastatic activities of fullerenol in a mouse breast cancer model. Carbon 48: 2231–2243. doi: 10.1016/j.carbon.2010.02.032
![]() |
[35] |
Xu J-Y, Su Y-Y, Cheng J-S, et al. (2010) Protective effects of fullerenol on carbon tetrachloride-induced acute hepatotoxicity and nephrotoxicity in rats. Carbon 48: 1388–1396. doi: 10.1016/j.carbon.2009.12.029
![]() |
[36] |
Mikawa M, Kato H, Okumura M, et al. (2001) Paramagnetic Water-Soluble Metallofullerenes Having the Highest Relaxivity for MRI Contrast Agents. Bioconjugate Chem 12: 510–514. doi: 10.1021/bc000136m
![]() |
[37] |
Chen C, Xing G, Wang J, et al. (2005) Multihydroxylated [Gd@C82(OH)22]n Nanoparticles: Antineoplastic Activity of High Efficiency and Low Toxicity. Nano Lett 5: 2050–2057. doi: 10.1021/nl051624b
![]() |
[38] |
Aoshima H, Kokubo K, Shirakawa S, et al. (2009) Antimicrobial activity of fullerenes and their hydroxylated derivatives. Biocontrol Sci 14: 69–72. doi: 10.4265/bio.14.69
![]() |
[39] |
Guldi DM, Asmus K-D (1999) Activity of water-soluble fullerenes towards ·OH-radicals and molecular oxygen. Radiat Phys Chem 56: 449–456. doi: 10.1016/S0969-806X(99)00325-4
![]() |
[40] |
Lai HS, Chen WJ, Chiang LY (2000) Free radical scavenging activity of fullerenol on the ischemia-reperfusion intestine in dogs. World J Surg 24: 450–454. doi: 10.1007/s002689910071
![]() |
[41] |
Sun D, Zhu Y, Liu Z, et al. (1997) Active oxygen radical scavenging ability of water-soluble fullerenols. Chin Sci Bull 42: 748–752. doi: 10.1007/BF03186969
![]() |
[42] |
Dugan LL, Gabrielsen JK, Yu SP, et al. (1996) Buckminsterfullerenol free radical scavengers reduce excitotoxic and apoptotic death of cultured cortical neurons. Neurobiol Dis 3: 129–135. doi: 10.1006/nbdi.1996.0013
![]() |
[43] | Chiang LY, Lu F-J, Lin J-T (1995) Free radical scavenging activity of water-soluble fullerenols. J Chem Soc, Chem Commun: 1283–1284. |
[44] |
Xiao L, Takada H, Maeda K, et al. (2005) Antioxidant effects of water-soluble fullerene derivatives against ultraviolet ray or peroxylipid through their action of scavenging the reactive oxygen species in human skin keratinocytes. Biomed Pharmacother 59: 351–358. doi: 10.1016/j.biopha.2005.02.004
![]() |
[45] |
Oberdorster E (2004) Manufactured nanomaterials (fullerenes, C60) induce oxidative stress in the brain of juvenile largemouth bass. Environ Health Perspect 112: 1058–1062. doi: 10.1289/ehp.7021
![]() |
[46] | Hamano T, Okuda K, Mashino T, et al. (1997) Singlet oxygen production from fullerene derivatives: effect of sequential functionalization of the fullerene core. Chem Commun 21–22. |
[47] |
Guldi DM, Prato M (2000) Excited-State Properties of C60 Fullerene Derivatives. Acc Chem Res 33: 695–703. doi: 10.1021/ar990144m
![]() |
[48] |
Jensen AW, Daniels C (2003) Fullerene-Coated Beads as Reusable Catalysts. J Org Chem 68: 207–210. doi: 10.1021/jo025926z
![]() |
[49] |
Jensen AW, Maru BS, Zhang X, et al. (2005) Preparation of fullerene-shell dendrimer-core nanoconjugates. Nano Lett 5: 1171–1173. doi: 10.1021/nl0502975
![]() |
[50] |
Foote CS (1994) Photophysical and photochemical properties of fullerenes. Top Curr Chem 169: 347–363. doi: 10.1007/3-540-57565-0_80
![]() |
[51] |
McCluskey DM, Smith TN, Madasu PK, et al. (2009) Evidence for Singlet-Oxygen Generation and Biocidal Activity in Photoresponsive Metallic Nitride Fullerene-Polymer Adhesive Films. ACS Appl Mater Interfaces 1: 882–887. doi: 10.1021/am900008v
![]() |
[52] | Alberti MN, Orfanopoulos M (2010) Recent mechanistic insights in the singlet oxygen ene reaction. Synlett 999–1026. |
[53] | Foote CS, Wexler S, Ando W (1965) Singlet oxygen. III. Product selectivity. Tetrahedron Lett 4111–4118. |
[54] | Dallas P, Rogers G, Reid B, et al. (2016) Charge separated states and singlet oxygen generation of mono and bis adducts of C60 and C70. Chem Phys 465–466: 28–39. |
[55] | Yano S, Naemura M, Toshimitsu A, et al. (2015) Efficient singlet oxygen generation from sugar pendant C60 derivatives for photodynamic therapy [Erratum to document cited in CA163:618143]. Chem Commun 51: 17631–17632. |
[56] |
Prat F, Stackow R, Bernstein R, et al. (1999) Triplet-State Properties and Singlet Oxygen Generation in a Homologous Series of Functionalized Fullerene Derivatives. J Phys Chem A 103: 7230–7235. doi: 10.1021/jp991237o
![]() |
[57] |
Tegos GP, Demidova TN, Arcila-Lopez D, et al. (2005) Cationic Fullerenes Are Effective and Selective Antimicrobial Photosensitizers. Chem Biol 12: 1127–1135. doi: 10.1016/j.chembiol.2005.08.014
![]() |
[58] |
Schinazi RF, Sijbesma R, Srdanov G, et al. (1993) Synthesis and virucidal activity of a water-soluble, configurationally stable, derivatized C60 fullerene. Antimicrob Agents Chemother 37: 1707–1710. doi: 10.1128/AAC.37.8.1707
![]() |
[59] | Dai L (1999) Advanced syntheses and microfabrications of conjugated polymers, C60-containing polymers and carbon nanotubes for optoelectronic applications. Polym Adv Technol 10: 357–420. |
[60] |
Phillips JP, Deng X, Todd ML, et al. (2008) Singlet oxygen generation and adhesive loss in stimuli-responsive, fullerene-polymer blends, containing polystyrene-block-polybutadiene- block-polystyrene and polystyrene-block-polyisoprene-block-polystyrene rubber-based adhesives. J Appl Polym Sci 109: 2895–2904. doi: 10.1002/app.28337
![]() |
[61] |
Lundin JG, Giles SL, Cozzens RF, et al. (2014) Self-cleaning photocatalytic polyurethane coatings containing modified C60 fullerene additives. Coatings 4: 614–629. doi: 10.3390/coatings4030614
![]() |
[62] |
Phillips JP, Deng X, Stephen RR, et al. (2007) Nano- and bulk-tack adhesive properties of stimuli-responsive, fullerene-polymer blends, containing polystyrene-block-polybutadiene- block-polystyrene and polystyrene-block-polyisoprene-block-polystyrene rubber-based adhesives. Polymer 48: 6773–6781. doi: 10.1016/j.polymer.2007.08.050
![]() |
[63] |
Samulski ET, DeSimone JM, Hunt MO, Jr., et al. (1992) Flagellenes: nanophase-separated, polymer-substituted fullerenes. Chem Mater 4: 1153–1157. doi: 10.1021/cm00024a011
![]() |
[64] |
Chiang LY, Wang LY, Kuo C-S (1995) Polyhydroxylated C60 Cross-Linked Polyurethanes. Macromolecules 28: 7574–7576. doi: 10.1021/ma00126a042
![]() |
[65] | Ahmed HM, Hassan MK, Mauritz KA, et al. (2014) Dielectric properties of C60 and Sc3N@C80 fullerenol containing polyurethane nanocomposites. J Appl Polym Sci 131: 40577–40588. |
[66] | Kokubo K, Takahashi R, Kato M, et al. (2014) Thermal and thermo-oxidative stability of thermoplastic polymer nanocomposites with arylated [60]fullerene derivatives. Polym Compos: 1–9. |
[67] |
Shin J, Nazarenko S, Phillips JP, et al. (2009) Physical and chemical modifications of thiol-ene networks to control activation energy of enthalpy relaxation. Polymer 50: 6281–6286. doi: 10.1016/j.polymer.2009.10.053
![]() |
[68] |
Hoyle CE, Bowman CN (2010) Thiol-ene click chemistry. Angew Chem Int Ed 49: 1540–1573. doi: 10.1002/anie.200903924
![]() |
[69] |
Hoyle CE, Lee TY, Roper T (2004) Thiol–enes: Chemistry of the past with promise for the future. J Polym Sci A Polym Chem 42: 5301–5338. doi: 10.1002/pola.20366
![]() |
[70] |
Cramer NB, Scott JP, Bowman CN (2002) Photopolymerizations of Thiol-Ene Polymers without Photoinitiators. Macromolecules 35: 5361–5365. doi: 10.1021/ma0200672
![]() |
[71] |
Li Q, Zhou H, Hoyle CE (2009) The effect of thiol and ene structures on thiol–ene networks: Photopolymerization, physical, mechanical and optical properties. Polymer 50: 2237–2245. doi: 10.1016/j.polymer.2009.03.026
![]() |
[72] |
Northrop BH, Coffey RN (2012) Thiol-Ene Click Chemistry: Computational and Kinetic Analysis of the Influence of Alkene Functionality. J Am Chem Soc 134: 13804–13817. doi: 10.1021/ja305441d
![]() |
[73] |
Singh R, Goswami T (2011) Understanding of thermo-gravimetric analysis to calculate number of addends in multifunctional hemi-ortho ester derivatives of fullerenol. Thermochimica Acta 513: 60–67. doi: 10.1016/j.tca.2010.11.012
![]() |
[74] |
Barker EM, Buchanan JP (2016) Thiol-ene polymer microbeads prepared under high-shear and their successful utility as a heterogeneous photocatalyst via C60-capping. Polymer 92: 66–73. doi: 10.1016/j.polymer.2016.03.091
![]() |
[75] | Jockusch S, Turro NJ (1998) Phosphinoyl Radicals: Structure and Reactivity. A Laser Flash Photolysis and Time-Resolved ESR Investigation. J Am Chem Soc 120: 11773–11777. |
[76] |
Ruoff RS, Tse DS, Malhotra R, et al. (1993) Solubility of fullerene (C60) in a variety of solvents. J Phys Chem 97: 3379–3383. doi: 10.1021/j100115a049
![]() |
[77] | Ginzburg BM, Shibaev LA, Melenevskaja EY, et al. (2004) Thermal and Tribological Properties of Fullerene-Containing Composite Systems. Part 1. Thermal Stability of Fullerene-Polymer Systems. J Macromol Sci Phys 43: 1193–1230. |
[78] |
Leifer SD, Goodwin DG, Anderson MS, et al. (1995) Thermal decomposition of a fullerene mix. Phys Rev B Condens Matter 51: 9973–9981. doi: 10.1103/PhysRevB.51.9973
![]() |
[79] | Mackey MA (2011) Exploration in metallic nitride fullerenes and oxometallic fullerenes: A new class of metallofullerenes [Ph.D. Dissertation]. Hattiesburg, MS: The University of Southern Mississippi. |
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