Research article Special Issues

The role of impurities in the shape, structure and physical properties of semiconducting oxide nanostructures grown by thermal evaporation

  • A thermal evaporation method developed in the research group enables to grow and design several morphologies of semiconducting oxide nanostructures, such as Ga2O3, GeO2 or Sb2O3, among others, and some ternary oxide compounds (ZnGa2O4, Zn2GeO4). In order to tailor physical properties, a successful doping of these nanostructures is required. However, for nanostructured materials, doping may affect not only their physical properties, but also their morphology during the thermal growth process. In this paper, we will show some examples of how the addition of impurities may result into the formation of complex structures, or changes in the structural phase of the material. In particular, we will consider the addition of Sn and Cr impurities into the precursors used to grow Ga2O3, Zn2GeO4 and Sb2O3 nanowires, nanorods or complex nanostructures, such as crossing wires or hierarchical structures. Structural and optical properties were assessed by electron microscopy (SEM and TEM), confocal microscopy, spatially resolved cathodoluminescence (CL), photoluminescence, and Raman spectroscopies. The growth mechanisms, the luminescence bands and the optical confinement in the obtained oxide nanostructures will be discussed. In particular, some of these nanostructures have been found to be of interest as optical microcavities. These nanomaterials may have applications in optical sensing and energy devices.

    Citation: Iñaki López, Teresa Cebriano, Pedro Hidalgo, Emilio Nogales, Javier Piqueras, Bianchi Méndez. The role of impurities in the shape, structure and physical properties of semiconducting oxide nanostructures grown by thermal evaporation[J]. AIMS Materials Science, 2016, 3(2): 425-433. doi: 10.3934/matersci.2016.2.425

    Related Papers:

    [1] Vittoria Raimondi, Alessandro Grinzato . A basic introduction to single particles cryo-electron microscopy. AIMS Biophysics, 2022, 9(1): 5-20. doi: 10.3934/biophy.2022002
    [2] Joshua Holcomb, Nicholas Spellmon, Yingxue Zhang, Maysaa Doughan, Chunying Li, Zhe Yang . Protein crystallization: Eluding the bottleneck of X-ray crystallography. AIMS Biophysics, 2017, 4(4): 557-575. doi: 10.3934/biophy.2017.4.557
    [3] Stephanie H. DeLuca, Samuel L. DeLuca, Andrew Leaver-Fay, Jens Meiler . RosettaTMH: a method for membrane protein structure elucidation combining EPR distance restraints with assembly of transmembrane helices. AIMS Biophysics, 2016, 3(1): 1-26. doi: 10.3934/biophy.2016.1.1
    [4] Adam Redzej, Gabriel Waksman, Elena V Orlova . Structural studies of T4S systems by electron microscopy. AIMS Biophysics, 2015, 2(2): 184-199. doi: 10.3934/biophy.2015.2.184
    [5] Riyaz A. Mir, Jeff Lovelace, Nicholas P. Schafer, Peter D. Simone, Admir Kellezi, Carol Kolar, Gaelle Spagnol, Paul L. Sorgen, Hamid Band, Vimla Band, Gloria E. O. Borgstahl . Biophysical characterization and modeling of human Ecdysoneless (ECD) protein supports a scaffolding function. AIMS Biophysics, 2016, 3(1): 195-210. doi: 10.3934/biophy.2016.1.195
    [6] Angel Rivera-Calzada, Andrés López-Perrote, Roberto Melero, Jasminka Boskovic, Hugo Muñoz-Hernández, Fabrizio Martino, Oscar Llorca . Structure and Assembly of the PI3K-like Protein Kinases (PIKKs) Revealed by Electron Microscopy. AIMS Biophysics, 2015, 2(2): 36-57. doi: 10.3934/biophy.2015.2.36
    [7] Wei Zhang, Sheng Cao, Jessica L. Martin, Joachim D. Mueller, Louis M. Mansky . Morphology and ultrastructure of retrovirus particles. AIMS Biophysics, 2015, 2(3): 343-369. doi: 10.3934/biophy.2015.3.343
    [8] Jany Dandurand, Angela Ostuni, Maria Francesca Armentano, Maria Antonietta Crudele, Vincenza Dolce, Federica Marra, Valérie Samouillan, Faustino Bisaccia . Calorimetry and FTIR reveal the ability of URG7 protein to modify the aggregation state of both cell lysate and amylogenic α-synuclein. AIMS Biophysics, 2020, 7(3): 189-203. doi: 10.3934/biophy.2020015
    [9] Ta-Chou Huang, Wolfgang B. Fischer . Sequence–function correlation of the transmembrane domains in NS4B of HCV using a computational approach. AIMS Biophysics, 2021, 8(2): 165-181. doi: 10.3934/biophy.2021013
    [10] Davide Sala, Andrea Giachetti, Antonio Rosato . Molecular dynamics simulations of metalloproteins: A folding study of rubredoxin from Pyrococcus furiosus. AIMS Biophysics, 2018, 5(1): 77-96. doi: 10.3934/biophy.2018.1.77
  • A thermal evaporation method developed in the research group enables to grow and design several morphologies of semiconducting oxide nanostructures, such as Ga2O3, GeO2 or Sb2O3, among others, and some ternary oxide compounds (ZnGa2O4, Zn2GeO4). In order to tailor physical properties, a successful doping of these nanostructures is required. However, for nanostructured materials, doping may affect not only their physical properties, but also their morphology during the thermal growth process. In this paper, we will show some examples of how the addition of impurities may result into the formation of complex structures, or changes in the structural phase of the material. In particular, we will consider the addition of Sn and Cr impurities into the precursors used to grow Ga2O3, Zn2GeO4 and Sb2O3 nanowires, nanorods or complex nanostructures, such as crossing wires or hierarchical structures. Structural and optical properties were assessed by electron microscopy (SEM and TEM), confocal microscopy, spatially resolved cathodoluminescence (CL), photoluminescence, and Raman spectroscopies. The growth mechanisms, the luminescence bands and the optical confinement in the obtained oxide nanostructures will be discussed. In particular, some of these nanostructures have been found to be of interest as optical microcavities. These nanomaterials may have applications in optical sensing and energy devices.


    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 $ \mathcal{X}:\mathcal{C}\subseteq\mathbb{R}\to\mathbb{R} $ is said to be $ AA $-convex, if

    $ (1-\mu)\mathcal{X}(x)+\mu\mathcal{X}(y)\geq \mathcal{X}((1-\mu)x+ty),\quad\forall x,y\in\mathcal{C},\mu\in[0,1], $

    where $ \mathcal{C} $ is a convex set.

    Definition 1.2 ([18]). ($ GG $-convex functions) A function $ \mathcal{X}:\mathcal{G}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ is said to be $ GG $-convex, if

    $ X1μ(x)Xμ(y)X(x1μyμ),x,yG,μ[0,1],
    $

    where $ \mathcal{G} $ is a geometric convex set.

    Definition 1.3 ([13]). ($ HH $-convex functions) A function $ \mathcal{X}:\mathcal{H}\subseteq\mathbb{R}_{+}\to\mathbb{R} $ is said to be $ HH $-convex, if

    $ X(x)X(y)μX(x)+(1μ)X(y)X(xy(1μ)x+ty),x,yH,μ[0,1],
    $

    where $ \mathcal{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 $ \mathcal{X}:I\subseteq\mathbb{R}\to\mathbb{R} $ be a convex function, then

    $ X(c+d2)1dcdcX(x)dxX(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 $ \mathscr{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 $ \mathscr{M} $-convex functions, $ \log $-$ \mathscr{M} $-convex and quasi $ \mathscr{M} $-convex functions. First of all for the sake of simplicity, we take $ \mathcal{G} = \sqrt{cd} $ and $ \mathcal{A} = \frac{c+d}{2} $.

    Definition 2.1. A function $ \mathcal{X}:D\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ is said to be $ \mathscr{M} $-convex function, if

    $ X((1μ)G+μA)(1μ)X(G)+tX(A),c,dD,μ[0,1].
    $

    Definition 2.2. A function $ \mathcal{X}:D\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ is said to be $ \log $-$ \mathscr{M} $-convex function, if

    $ X((1μ)G+μA)X1μ(G)Xμ(A),c,dD,μ[0,1].
    $

    Definition 2.3. A function $ \mathcal{X}:D\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ is said to be quasi $ \mathscr{M} $-convex function, if

    $ X((1μ)G+μA)max{X(G),X(A)},c,dD,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 $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a differentiable function on $ I^{\circ} $, $ c, d\in I^{\circ} $ with $ c < d $. If $ \mathcal{X}'\in L[c, d] $, then

    $ X(G)+X(A)22(dc)2AGX(x)dx=(dc)2410(12μ)X(μG+(1μ)A)dμ.
    $

    Proof. It suffices to show that

    $ 10(12μ)X(μG+(1μ)A)dμ=2X(G)+X(A)(dc)28(dc)4AGX(x)dx.
    $

    This implies

    $ (dc)2410(12μ)X(μG+(1μ)A)dμ=X(G)+X(A)22(dc)2AGX(x)dx.
    $

    This completes the proof.

    Now utilizing Lemma 3.1, we derive our next results.

    Theorem 3.2. Let $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a differentiable function on $ I^{\circ} $, $ c, d\in I^{\circ} $ with $ c < d $ and $ \mathcal{X}'\in L[c, d] $. If $ |\mathcal{X}'| $ is $ \mathscr{M} $-convex function, then

    $ |X(G)+X(A)22(dc)2AGX(x)dx|(dc)216[|X(G)|+|X(A)|].
    $

    Proof. Using Lemma 3.1, property of the modulus and the fact that $ |\mathcal{X}'| $ is $ \mathscr{M} $-convex function, we have

    $ |X(G)+X(A)22(dc)2AGX(x)dx|(dc)2410|12μ||X(μG+(1μ)A)|dμ(dc)2410|12μ|[μ|X(G)|+(1μ)|X(A)|]dμ=(dc)216[|X(G)|+|X(A)|].
    $

    This completes the proof.

    If we apply Theorem 3.2 for $ \log $-$ \mathscr{M} $-convex functions, then

    Theorem 3.3. Let $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a differentiable function on $ I^{\circ} $, $ c, d\in I^{\circ} $ with $ c < d $ and $ \mathcal{X}'\in L[c, d] $, If $ |\mathcal{X}'| $ is decreasing and $ \log $-$ \mathscr{M} $-convex function, then

    $ |X(G)+X(A)22(dc)2AGX(x)dx|(dc)24[2+4w2wlogw+wlogwlogw2],
    $

    where $ w = \frac{|\mathcal{X}'(\mathcal{G})|}{\big|\mathcal{X}'(\mathcal{A})\big|} $.

    Theorem 3.4. Let $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a differentiable function on $ I^{\circ} $, $ c, d\in I^{\circ} $ with $ c < d $ and $ \mathcal{X}'\in L[c, d] $. If $ |\mathcal{X}'|^{q} $, where $ \frac{1}{p}+\frac{1}{q} = 1 $ is $ \mathscr{M} $-convex function, then

    $ |X(G)+X(A)22(dc)2AGX(x)dx|(dc)24(1p+1)1p(|X(G)|q+|X(A)|q2)1q.
    $

    Proof. Using Lemma 3.1, Holder's inequality and the fact that $ |\mathcal{X}'|^{q} $ is $ \mathscr{M} $-convex functions, we have

    $ |X(G)+X(A)22(dc)2AGX(x)dx|(dc)24(10|12μ|pdμ)1p(10|X(μG+(1μ)A)|dμ)1q(dc)24(1p+1)1p(10[μ|X(G)|q+(1μ)|X(A)|q]dμ)1q=(dc)24(1p+1)1p(|X(G)|q+|X(A)|q2)1q.
    $

    This completes the proof.

    Theorem 3.5. Let $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a differentiable function on $ I^{\circ} $, $ c, d\in I^{\circ} $ with $ c < d $ and $ \mathcal{X}'\in L[c, d] $. If $ |\mathcal{X}'|^{q} $, where $ q\geq1 $ is $ \mathscr{M} $-convex function, then

    $ |X(G)+X(A)22(dc)2AGX(x)dx|(dc)28(|X(G)|q+|X(A)|q2)1q.
    $

    Proof. Using Lemma 3.1, power mean inequality and the fact that $ |\mathcal{X}'| $ is $ \mathscr{M} $-convex functions, we have

    $ |X(G)+X(A)22(dc)2AGX(x)dx|(dc)24(10|12μ|dμ)11q(10|12μ||X(μG+(1μ)A)|dμ)1q(dc)24(12)11q(10|12μ|[μ|X(G)|q+(1μ)|X(A)|q]dμ)1q=(dc)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 $ \mathscr{M} $-convex functions. Before that we recall basic definition of Riemann-Liouville fractional integrals.

    Definition 4.1 ([15]). Let $ {\mathcal{X}}\in L[c, d] $, where $ c\geq 0. $ The Riemann-Liouville integrals $ J_{c+}^{\nu }{\mathcal{X}} $ and $ J_{d-}^{\nu }{\mathcal{X}}, $ of order $ \nu > 0, $ are defined by

    $ Jνc+X(x)=1Γ(ν)xc(xμ)ν1X(μ)dμ, for x>c
    $

    and

    $ JνdX(x)=1Γ(ν)dx(μx)ν1X(μ)dμ, for x<d,
    $

    respectively. Here, $ \Gamma (\nu) = \int_{0}^{\infty }e^{-\mu}\mu^{\nu -1} \mathrm{d}\mu $ is the Gamma function. We also make the convention

    $ J0c+X(x)=J0dX(x)=X(x).
    $

    We now derive a new auxiliary result utilizing the definition of Riemann-Liouville fractional integrals.

    Lemma 4.1. Let $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a differentiable function. If $ \mathcal{X}'\in L[c, d] $, then

    $ X(G)+X(A)22α1Γ(α+1)(dc)2[Jα(A)X(G)+Jα(G)+X(A)]=(dc)2410[(1μ)αμα]X(μG+(1μ)A)dμ.
    $

    Proof. It suffices to show that

    $ I=10[(1μ)αμα]X(μG+(1μ)A)dμ=10(1μ)αX(μG+(1μ)A)dμ10μαX(μG+(1μ)A)dμ=I1I2.
    $
    (4.1)

    Now using change of variable technique and definition of Riemann-Liouville fractional integrals, we have

    $ I1=10(1μ)αX(μG+(1μ)A)dμ=2(dc)2X(A)2α+1Γ(α+1)(dc)2(α+1)1Γ(α)AG(xG)α1X(x)dx=2(dc)2X(A)2α+1Γ(α+1)(dc)2(α+1)Jα(A)X(G).
    $
    (4.2)

    Similarly

    $ I2=10μαX(μG+(1μ)A)dμ=2(dc)2X(G)+2α+1Γ(α+1)(dc)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 $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a differentiable function and $ \mathcal{X}'\in L[c, d] $. If $ |\mathcal{X}'| $ is $ \mathscr{M} $-convex function, then

    $ |X(G)+X(A)22α1Γ(α+1)(dc)2[Jα(A)X(G)+Jα(G)+X(A)]|(dc)24(α+1)(112α)[|X(a)|+|X(b)|].
    $

    Proof. Using Lemma 4.1 and the property of modulus, we have

    $ |X(G)+X(A)22α1Γ(α+1)(dc)2[Jα(A)X(G)+Jα(G)+X(A)]|10(dc)24|(1μ)αμα||X(μG+(1μ)A)|dμ.
    $

    Since it is given that $ |\mathcal{X}'| $ is $ \mathscr{M} $-convex function, so we have

    $ |X(G)+X(A)22α1Γ(α+1)(dc)2[Jα(A)X(G)+Jα(G)+X(A)]|10(dc)24|(1μ)αμα|[μ|X(G)|+(1μ)|X(A)|]dμ=(dc)24[|X(G)|10μ|(1μ)αμα|dμ+|X(A)|10(1μ)|(1μ)αμα|dμ]=(dc)24(α+1)(112α)[|X(a)|+|X(b)|].
    $

    This completes the proof.

    In this section, we derive some quantum analogues of Hermite-Hadamard like inequalities using $ \mathscr{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 $ \mathcal{X}:I\subset\mathbb{R}\rightarrow\mathbb{R} $ be a continuous function. Then $ q $-integral on $ I $ is defined as:

    $ xaX(μ)adqμ=(1q)(xa)n=0qnX(qnx+(1qn)a),
    $
    (5.1)

    for $ x\in J $.

    The following result will play significant role in main results of the section.

    Lemma 5.1 ([25]). Let $ \alpha\in\mathbb{R}\setminus\{-1\} $, then

    $ xa(μa)αadqμ=(1q1qα+1)(xa)α+1.
    $

    Lemma 5.2. Let $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a $ q $-differentiable function on $ I^{\circ} $, $ c, d\in I^{\circ} $ with $ c < d $. If $ \mathrm{D}_{q}\mathcal{X} $ is an integrable function with $ 0 < q < 1 $, then

    $ 2(dc)2AGX(μ)dqμqf(G)+X(A)1+q=q(dc)22(1+q)10(1(1+q)μ)Dq((1μ)G+μA)dqμ.
    $

    Proof. It suffices to show that

    $ 10(1(1+q)μ)Dq((1μ)G+μA)dqμ=2(dc)210(X((1μ)G+μA)X((1qμ)G+qμA)(1q)μ)dqμ2(1+q)(dc)210μ(X((1μ)G+μA)X((1qμ)G+qμA)(1q)μ)dqμ=2(dc)2[n=0X((1qn)G+qnA)n=0X((1qn+1)G+qn+1A)]2(1+q)(dc)2[n=0qnX((1qn)G+qnA)n=0qnX((1qn+1)G+qn+1A)]=2(dc)2[X(A)X(G)]2(1+q)(dc)2n=0qnX((1qn)G+qnA)+2(1+q)q(dc)2n=1qnX((1qn)G+qnA)=2(dc)2[X(A)X(G)]2(1+q)(dc)2n=0qnX((1qn)G+qnA)+2(1+q)q(dc)2[X(A)X(A)+n=1qnX((1qn)G+qnA)]=2q(dc)2[qf(G)+X(A)]+4(1+q)q(dc)4AGX(μ)dqμ.
    $

    This completes the proof.

    Now using Lemma 5.2, we derive our next results.

    Theorem 5.3. Let $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a $ q $-differentiable function on $ I^{\circ} $, $ c, d\in I^{\circ} $ with $ c < d $ and $ \mathrm{D}_{q}\mathcal{X} $ is an integrable function with $ 0 < q < 1 $. If $ |\mathrm{D}_{q}\mathcal{X}| $ is $ \mathscr{M} $-convex, then

    $ |2(dc)2AGX(μ)dqμqf(G)+X(A)1+q|q(dc)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(dc)2AGX(μ)dqμqf(G)+X(A)1+q|=|q(dc)22(1+q)10(1(1+q)μ)DqX((1μ)G+μA)dqμ|q(dc)22(1+q)10|1(1+q)μ|[(1μ)|DqX(G)|+μDq|DqX(A)|]dqμ=q(dc)22(1+q){|DqX(G)|10(1μ)|1(1+q)μ|dqμ+|DqX(A)|10μ|1(1+q)μ|dqμ}=q(dc)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 $ \mathcal{X}:I^{\circ}\subseteq\mathbb{R}_{+}\to\mathbb{R}_{+} $ be a $ q $-differentiable function on $ I^{\circ} $, $ c, d\in I^{\circ} $ with $ c < d $ and $ \mathrm{D}_{q}\mathcal{X} $ is an integrable function with $ 0 < q < 1 $. If $ |\mathrm{D}_{q}\mathcal{X}|^{r} $ is $ \mathscr{M} $-convex, where $ r > 1 $, then

    $ |2(dc)2AGX(μ)dqμqf(G)+X(A)1+q|q(dc)22(1+q)(2q(1+q)2)11r(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(dc)2AGX(μ)dqμqf(G)+X(A)1+q|=|q(dc)22(1+q)10(1(1+q)μ)DqX((1μ)G+μA)dqμ|q(dc)22(1+q)(10|1(1+q)μ|dqμ)11r×(10|1(1+q)μ|[(1μ)|DqX(G)|r+μ|DqX(A)|r]dqμ)1r=q(dc)22(1+q)(2q(1+q)2)11r(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 $ \mathscr{M} $-convex functions, $ \log $-$ \mathscr{M} $-convex functions and quasi $ \mathscr{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 $ \mathscr{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] Devan RS, Patil RA, Lin J-H, et al. (2012) One-Dimensional Metal-Oxide Nanostructures: Recent Developments in Synthesis, Characterization, and Applications. Adv Func Mater 22: 3326–3370. doi: 10.1002/adfm.201201008
    [2] Chen X, Wong CKY, Yuan CA, et al. (2013) Nanowire-based gas sensors. Sens Actuators B Chem 177: 178–195.
    [3] Minami T (2005) Transparent conducting oxide semiconductors for transparent electrodes. Semicond Sci Technol 20: S35–S44. doi: 10.1088/0268-1242/20/4/004
    [4] Pan ZW, Dai ZR, Wang ZL (2001) Nanobelts of Semiconducting Oxides. Science 291: 1947–1949. doi: 10.1126/science.1058120
    [5] Liu B, Zeng H-C (2003) Hydrothermal Synthesis of ZnO Nanorods in the Diameter Regime of 50 nm. J Am Chem Soc 125: 4430–4431. doi: 10.1021/ja0299452
    [6] Available from: http://www.finegroup.es
    [7] Lorenz MR, Woods JF, Gambino RJ (1967) Some electrical properties of the semiconductor β-Ga2O3. J Phys Chem Solids 28: 403–404. doi: 10.1016/0022-3697(67)90305-8
    [8] Binet L, Gourier D (1998) Origin of the blue luminescence of β-Ga2O3. J Phys Chem Solids 59: 1241–1249. doi: 10.1016/S0022-3697(98)00047-X
    [9] Chin HS, Cheong KY, Razak KA (2010) Review on oxides of antimony nanoparticles: synthesis, properties, and applications. J Mater Sci 45: 5993–6008.
    [10] Ormand RG, Holland D (2007) Thermal phase transitions in antimony (III) oxides. J Solid State Chem 180: 2587–2596. doi: 10.1016/j.jssc.2007.07.004
    [11] Mizoguchi H, Kamiya T, Matsuishi S, et al. (2011) A germanate transparent conductive oxide. Nat Commun 2: 470.
    [12] Maximenko SI, Mazeina L, Picard YN, et al. (2009) Cathodoluminescence studies of the inhomogeneities in Sn-doped Ga2O3 nanowires. Nano Lett 9: 3245–3251. doi: 10.1021/nl901514k
    [13] López I, Castaldini A, Cavallini A, et al. (2014) β-Ga2O3 nanowires for ultraviolet light selective frequency photodetector. J Phys D Appl Phys 47: 415101. doi: 10.1088/0022-3727/47/41/415101
    [14] López I, Nogales E, Méndez B, et al. (2013) Influence of Sn and Cr Doping on Morphology and Luminescence of Thermally Grown Ga2O3 Nanowires. J Phys Chem C 117: 3036–3045. doi: 10.1021/jp3093989
    [15] Martínez-Criado G, Segura-Ruiz J, Chu M-H, et al. (2014) Crossed Ga2O3/SnO2 multiwire architecture: a local structure study with nanometer resolution. Nano Lett 14: 5479–5487. doi: 10.1021/nl502156h
    [16] Cebriano T, Méndez B, Piqueras J (2012) Study of luminescence and optical resonances in Sb2O3 micro- and nanotriangles. J Nanopart Res 14: 1215. doi: 10.1007/s11051-012-1215-8
    [17] Cebriano T, Méndez B, Piqueras J (2013) Sb2O3 microrods: self-assembly phenomena, luminescence and phase transition. J Nanopart Res 15: 1667.
    [18] Cebriano T, Hidalgo P, Maestre D, et al. (2014) Study of mechanical resonances of Sb2O3 micro- and nanorods. Nanotechnol. 25: 235701. doi: 10.1088/0957-4484/25/23/235701
    [19] Hidalgo P, López A, Méndez B, et al. (2016) Synthesis and optical properties of Zn2GeO4 microrods. Acta Materialia 104: 84–90. doi: 10.1016/j.actamat.2015.11.023
    [20] Hidalgo P, Méndez B, Piqueras J (2008) Sn doped GeO2 nanowires with waveguiding behaviour. Nanotechnol 19: 455705.
    [21] Nogales E, García JA, Méndez B, et al. (2007) Doped gallium oxide nanowires with waveguiding behavior. Appl Phys Lett 91: 133108. doi: 10.1063/1.2790809
    [22] López I, Nogales E, Méndez B, et al. (2012) Resonant cavity modes in gallium oxide microwires. Appl Phys Lett 100: 261910. doi: 10.1063/1.4732153
    [23] Bartolome J, Cremades A, Piqueras A (2013) Thermal growth, luminescence and whispering gallery resonance modes of indium oxide microrods and microcrystals. J Mater Chem C 1: 6790–6799. doi: 10.1039/c3tc31195c
  • This article has been cited by:

    1. Xuekui Yu, Jonathan Jih, Jiansen Jiang, Z. Hong Zhou, Atomic structure of the human cytomegalovirus capsid with its securing tegument layer of pp150, 2017, 356, 0036-8075, eaam6892, 10.1126/science.aam6892
    2. Hua Jin, Yong-Liang Jiang, Feng Yang, Jun-Tao Zhang, Wei-Fang Li, Ke Zhou, Jue Ju, Yuxing Chen, Cong-Zhao Zhou, Capsid Structure of a Freshwater Cyanophage Siphoviridae Mic1, 2019, 27, 09692126, 1508, 10.1016/j.str.2019.07.003
    3. Joshua M. Hardy, Rhys A. Dunstan, Rhys Grinter, Matthew J. Belousoff, Jiawei Wang, Derek Pickard, Hariprasad Venugopal, Gordon Dougan, Trevor Lithgow, Fasséli Coulibaly, The architecture and stabilisation of flagellotropic tailed bacteriophages, 2020, 11, 2041-1723, 10.1038/s41467-020-17505-w
    4. Yanting Tang, An Mu, Yuying Zhang, Shan Zhou, Weiwei Wang, Yuezheng Lai, Xiaoting Zhou, Fengjiang Liu, Xiuna Yang, Hongri Gong, Quan Wang, Zihe Rao, Cryo-EM structure of Mycobacterium smegmatis DyP-loaded encapsulin, 2021, 118, 0027-8424, e2025658118, 10.1073/pnas.2025658118
    5. James M. Polson, Edgar J. Garcia, Alexander R. Klotz, Flatness and intrinsic curvature of linked-ring membranes, 2021, 17, 1744-683X, 10505, 10.1039/D1SM01307F
    6. Ning Cui, Feng Yang, Jun-Tao Zhang, Hui Sun, Yu Chen, Rong-Cheng Yu, Zhi-Peng Chen, Yong-Liang Jiang, Shu-Jing Han, Xudong Xu, Qiong Li, Cong-Zhao Zhou, Rebecca Ellis Dutch, Capsid Structure of Anabaena Cyanophage A-1(L) , 2021, 95, 0022-538X, 10.1128/JVI.01356-21
    7. Jing Zheng, Wenyuan Chen, Hao Xiao, Fan Yang, Xiaowu Li, Jingdong Song, Lingpeng Cheng, Hongrong Liu, A Capsid Structure of Ralstonia solanacearum podoviridae GP4 with a Triangulation Number T = 9, 2022, 14, 1999-4915, 2431, 10.3390/v14112431
    8. Jennifer M. Podgorski, Krista Freeman, Sophia Gosselin, Alexis Huet, James F. Conway, Mary Bird, John Grecco, Shreya Patel, Deborah Jacobs-Sera, Graham Hatfull, Johann Peter Gogarten, Janne Ravantti, Simon J. White, A structural dendrogram of the actinobacteriophage major capsid proteins provides important structural insights into the evolution of capsid stability, 2023, 31, 09692126, 282, 10.1016/j.str.2022.12.012
    9. Hao Pang, Fenxia Fan, Jing Zheng, Hao Xiao, Zhixue Tan, Jingdong Song, Biao Kan, Hongrong Liu, Three-dimensional structures of Vibrio cholerae typing podophage VP1 in two states, 2024, 09692126, 10.1016/j.str.2024.10.005
    10. Michael Woodson, Nikolai S. Prokhorov, Seth D. Scott, Wei Zhao, Wei Zhang, Kyung H. Choi, Paul J. Jardine, Marc C. Morais, Phi29 assembly intermediates reveal how scaffold interactions with capsid protein drive capsid construction and maturation, 2025, 11, 2375-2548, 10.1126/sciadv.adk8779
  • Reader Comments
  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(6018) PDF downloads(1194) Cited by(2)

Figures and Tables

Figures(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog