Coefficient estimates of some classes of rational functions

OMA-Vol. 2 (2018), Issue 2, pp. 114–128 | Open Access Full-Text PDF
Hanan Darwish, Suliman Sowileh, Abd AL-Monem Lashin
Abstract:Let \(\mathcal{A}\) be the class of analytic and univalent functions in the open unit disc \(\Delta\) normalized such that \(f(0)=0=f^{\prime }(0)-1.\) In this paper, for \(\psi \in \mathcal{A}\) of the form \(\frac{z}{f(z)}, f(z)=1+\sum\limits_{n=1}^{\infty }a_{_{n}}z^{n}\) and \(0\leq \alpha \leq 1,\) we introduce and investigate interesting subclasses \(\mathcal{H}_{\sigma }(\phi ), \;S_{\sigma }(\alpha ,\phi ), \; M_{\sigma }(\alpha ,\phi ),\) \( \Im _{\alpha} (\alpha ,\phi )\) and \(\beta _{\alpha}(\lambda ,\phi ) \left( \lambda \geq 0 \right)\) of analytic and bi-univalent Ma-Minda starlike and convex functions. Furthermore, we find estimates on the coefficients \(\left\vert a_{1}\right\vert\) and \(\left\vert a_{2}\right\vert\) for functions in these classess. Several related classes of functions are also considered.
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Asymptotic stability and blow-up of solutions for the generalized boussinesq equation with nonlinear boundary condition

OMA-Vol. 2 (2018), Issue 2, pp. 93–113 | Open Access Full-Text PDF
Jian Dang, Qingying Hu, Hongwei Zhang
Abstract:In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish both the existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.
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Super \((a,d)\)-\(C_3\)-antimagicness of a Corona Graph

OMS-Vol. 2 (2018), Issue 1, pp. 371–378 Open Access Full-Text PDF
Noshad Ali, Muhammad Awais Umar, Afshan Tabassum, Abdul Raheem
Abstract:A simple graph \(G=(V(G),E(G))\) admits an \(H\)-covering if \(\forall \ e \in E(G)\ \Rightarrow\ e \in E(H’)\) for some \((H’ \cong H )\subseteq G\). A graph \(G\) with \(H\) covering is an \((a,d)\)-\(H\)-antimagic if for bijection \(f:V\cup E \to \{1,2,\dots, |V(G)|+|E(G)| \}\), the sum of labels of all the edges and vertices belong to \(H’\) constitute an arithmetic progression \(\{a, a+d, \dots, a+(t-1)d\}\), where \(t\) is the number of subgraphs \(H’\). For \(f(V)= \{ 1,2,3,\dots,|V(G)|\}\), the graph \(G\) is said to be super \((a,d)\)-\(H\)-antimagic and for \(d=0\) it is called  \(H\)-supermagic. In this paper, we investigate the existence of super \((a,d)\)-\(C_3\)-antimagic labeling of a corona graph, for differences \(d=0,1,\dots, 5\).
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Fractional Integral Inequalities on Time Scales

OMS-Vol. 2 (2018), Issue 1, pp. 361–370 Open Access Full-Text PDF
Deniz Uçar, Veysel F. Hatipo\(\breve{\text{g}}\)lu, Aysegűl Akincali
Abstract:In this paper, we use the Delta Riemann-Liouville fractional integrals to establish some new integral inequalities for the Chebyshev functional in the case of two synchronous functions on time scales. Our results improve the inequalities for the discrete and continuous cases.
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Old symmetry problem revisited

OMA-Vol. 2 (2018), Issue 2, pp. 89–92 | Open Access Full-Text PDF
Alexander G. Ramm
Abstract:It is proved that if the problem \(\nabla^2u=1\) in \(D\), \(u|_S=0\), \(u_N=m:=|D|/|S|\) then \(D\) is a ball. There were at least two different proofs published of this result. The proof given in this paper is novel and short.
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Calculating Degree Based Topological Indices of Line Graph of \(HAC_{5}C_{6}C_{7}[p,q]\) Nanotube via M-Polynomial

OJC-Vol. 1 (2018), Issue 2, pp. 01–10 | Open Access Full-Text PDF
Aziz ur Rehman, Waseem Khalid
Abstract:The application of graph theory in chemical and molecular structure research far exceeds people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonded by edges. In this report, we study the M-polynomial of line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) and recover many degree-based topological indices from it.
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Zagreb Polynomials and redefined Zagreb indices of line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) Nanotube

OJC-Vol. 1 (2018), Issue 1, pp. 26–35 | Open Access Full-Text PDF
Aziz ur Rehman, Waseem Khalid
Abstract:The application of graph theory in chemical and molecular structure research far exceeds people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonded by edges. In this report, we study the Zagreb-polynomials of line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) and compute some degree-based topological indices from it.
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On Torsion and Finite Extension of \(FC\) and \(\tau N_{K}\) Groups in Certain Classes of Finitely Generated Groups

OMS-Vol. 2 (2018), Issue 1, pp. 351–360 Open Access Full-Text PDF
Mourad Chelgham, Mohamed Kerada
Abstract:Let \(k>0\) an integer. \(F\), \(\tau \), \(N\), \(N_{k}\), \(N_{k}^{(2)}\) and \(A\) denote the classes of finite, torsion, nilpotent, nilpotent of class at most \(k\), group which every two generator subgroup is \(N_{k}\) and abelian groups respectively. The main results of this paper is, firstly, we prove that, in the class of finitely generated \(\tau N\)-group (respectively \(FN\)-group) a \((FC)\tau \)-group (respectively \((FC)F\)-group) is a \(\tau A\)-group (respectively is \(FA\)-group). Secondly, we prove that a finitely generated \(\tau N\)-group (respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)\) (respectively \(((FN_{k})F,\infty)\)) is a \(\tau N_{k}^{(2)}\)-group (respectively \(FN_{k}^{(2)}\)-group). Thirdly we prove that a finitely generated \(\tau N\)-group ( respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)^{\ast}\) (respectively \(((FN_{k})F,\infty)^{\ast}\)) is a \(\tau N_{c}\)-group (respectively \(FN_{c}\)-group) for certain integer \(c\) and we extend this results to the class of \(NF\)-groups.
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Computing Reformulated First Zagreb Index of Some Chemical Graphs as an Application of Generalized Hierarchical Product of Graphs

OMS-Vol. 2 (2018), Issue 1, pp. 338–350 Open Access Full-Text PDF
Nilanjan De
Abstract:The generalized hierarchical product of graphs was introduced by L. Barrière et al. in 2009. In this paper, reformulated first Zagreb index of generalized hierarchical product of two connected graphs and hence as a special case cluster product of graphs are obtained. Finally using the derived results the reformulated first Zagreb index of some chemically important graphs such as square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer fullerene, zig-zag polyhex nanotube and dicentric dendrimers are computed.
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