majorization

Considering two special families of regular functions in an open unit disk based on quasi-subordination, we present sharp bounds for initial coefficient estimates and also determine the classical functional of Fekete-Szegö of functions in these families. Further, we discuss subordination and majorization results for the associated families. Few known and several new consequences are established.

For any simple graph $G$, the signless Laplacian matrix of $G$ is defined as $D(G)+A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of vertex degrees and the adjacency matrix of $G$, respectively. %Let $chi(G)$ be the chromatic number of $G$ Let $q(G)$ be the signless Laplacian spectral radius of $G$ (the largest eigenvalue of the signless Laplacian matrix of $G$). In this paper we find some relations between the chromatic number and the signless Laplacian spectral radius of graphs. In particular, we characterize all graphs $G$ of order $n$ with odd chromatic number $chi$ such that $q(G)=2nBig(1-frac{1}{chi}Big)$. Finally we show that if $G$ is a graph of order $n$ and with chromatic number $chi$, then under certain conditions, $q(G)<2nBig(1-frac{1}{chi}Big)-frac{2}{n}$. This result improves some previous similar results.

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A nonnegative real matrix is called ‎row‎ stochastic if sum of all entries in each row is one. For vectors ‎$‎x, y \in R_n‎$‎, it is said that ‎$‎x‎$‎ is left-right matrix majorized by ‎$‎y‎$ ‎and ‎write ‎‎$‎x\prec_{rl} y‎$‎ if for some row stochastic matrices ‎$‎A,B‎$‎; ‎$‎x=yA‎$ ‎and ‎‎$‎x‎^‎t=By^t‎$‎. A linear operator ‎$‎T : ‎\mathbb{R}_n‎\longrightarrow ‎‎\mathbb{R}_n‎$‎‎‎ is said to be a linear preserver of a given relation ‎$‎‎\mathcal{R}‎‎$‎ if ‎$‎x‎\mathcal{R}‎y‎$‎‎‎ implies that ‎$‎T(x)‎\mathcal{R}T(y)‎$‎‎‎.‎ In this parer we characterize the linear preservers of ‎$‎\prec_{rl}‎$‎ from ‎$‎\mathbb{R}_n‎$‎ to $‎\mathbb{R}_n‎$‎. ‎In fact, w‎e show that the linear preservers of ‎$‎\prec_{rl}‎$‎ from ‎$‎ mathbb{R}_n‎$‎ to $‎\mathbb{R}_n‎$ are the same as the linear preservers of ‎$‎\prec_{m}‎$‎ from ‎$‎\mathbb{R}_n‎$‎ to $‎\mathbb{R}_n‎$ for ‎$‎n\leq 3‎$‎; but for ‎$‎n\geq 4‎$‎; they are not the same.

In this paper, we consider the class of 4-convex functions and we obtain some inequalities related to 4-convex functions. Moreover, for k ≤ n, we present a majorization ≺k on Rn and we give some equivalent conditions for ≺4 on R4.

Let  MnMn be  the set of all nn-by-nn real  matrices, and let  RnRn be  the set of all nn-by-11 real (column) vectors. An nn-by-nn matrix R=[rij]R=[rij] with nonnegative entries is called row stochastic, if ∑nk=1rik∑k=1nrik is equal to 1 for all ii, (1≤i≤n)(1≤i≤n). In fact, Re=eRe=e, where e=(1,…,1)t∈Rne=(1,…,1)t∈Rn.  A matrix R∈MnR∈Mn  is called integral row stochastic, if each row has exactly one nonzero entry, +1+1, and other entries are zero.  In the present paper,  we provide an algorithm for constructing integral row stochastic matrices, and also we show the relationship between this algorithm and majorization theory.

Assume we have $k$ immediate (due)-annuities with different interest rates. Let ${bf i}=(i_1,i_2,...,i_k)$ and ${bf i^*}=(i^*_1,i^*_2,...,i^*_k)$ be two vectors of interest rates such that ${bf i^*}$ is majorized by ${bf i}$. It's shown that sum of present and accumulated value of annuities-immediate with interest rate ${bf i}$ is grater than sum of present value of annuities-immediate with interest rate ${bf i^*}$. We also prove the similar results for annuities-due.