Annals of the West University of Timisoara: Mathematics and Computer Science 

Abstract: Abstract In this paper, generalized Hirota Satsuma coupled KdV system is solved with tanh method and q-Homotopy analysis method. New fractional derivative definition called “conformable fractional derivative” used in the solution procedure. Tanh method with conformable derivative firstly introduced in the literature. By the graphics of analytical and approximate solutions, it is shown that, both methods provide an effective and powerful mathematical tool for solving nonlinear PDEs containing conformable fractional derivative.

Abstract: Abstract Our purpose in this paper is to consider a generalized form of the extended Hurwitz-Lerch Zeta function. For this extended Hurwitz-Lerch Zeta function, we obtain some classical properties which includes various integral representations, a differential formula, Mellin transforms and certain generating relations. We further consider an application to probability distributions and also point out some important special cases of the main results.

Abstract: Some inequalities of Hermite-Hadamard type for GA-convex functions de…ned on positive intervals are given. 1. Introduction Let I (0;1) be an interval; a real-valued function f : I ! R is said to be GA-convex (concave) on I if (1.1) f x y ( ) (1 ) f (x) + f (y) for all x; y 2 I and 2 [0; 1]. Since the condition (1.1) can be written as (1.2) f exp ((1 ) lnx+ ln y) ( ) (1 ) f exp (lnx) + f exp (ln y) ; then we observe that f : I ! R is GA-convex (concave) on I if and only if f exp is convex (concave) on ln I := fln z; z 2 Ig : If I = [a; b] then ln I = [ln a; ln b] : It is known that the function f (x) = ln (1 + x) is GA-convex on (0;1) [4]. For real and positive values of x, the Euler gamma function and its logarithmic derivative , the so-called digamma function, are de…ned by (x) := Z 1 0 t e dt and (x) := 0 (x) (x) : It has been shown in [54] that the function f : (0;1)! R de…ned by f (x) = (x) + 1 2x is GA-concave on (0;1) while the function g : (0;1)! R de…ned by g (x) = (x) + 1 2x + 1 12x2 is GA-convex on (0;1) : If [a; b] (0;1) and the function g : [ln a; ln b] ! R is convex (concave) on [ln a; ln b] ; then the function f : [a; b] ! R, f (t) = g (ln t) is GA-convex (concave) on [a; b] : Indeed, if x; y 2 [a; b] and 2 [0; 1] ; then f x y = g ln x y = g [(1 ) lnx+ ln y] ( ) (1 ) g (lnx) + g (ln y) = (1 ) f (x) + f (y) showing that f is GA-convex (concave) on [a; b] : 1991 Mathematics Subject Classi…cation. 26D15; 25D10. Key words and phrases. Convex functions, Integral inequalities, GA-Convex functions. 1 2 S. S. DRAGOMIR We recall that the classical Hermite-Hadamard inequality that states that (1.3) f a+ b 2 1 b a Z b a f (t) dt f (a) + f (b) 2 for any convex function f : [a; b]! R. For related results, see [1]-[20], [23]-[25], [26]-[35] and [36]-[46]. In [54] the authors obtained the following Hermite-Hadamard type inequality. Theorem 1. If b > a > 0 and f : [a; b]! R is a di¤erentiable GA-convex (concave) function on [a; b] ; then (1.4) f (I (a; b)) ( ) 1 b a Z b a f (t) dt ( ) b L (a; b) b a f (b)+ L (a; b) a b a f (a) : The identric mean I (a; b) is de…ned by I (a; b) := 1 e b aa 1 b a while the logarithmic mean is de…ned by L (a; b) := b a ln b ln a The di¤erentiability of the function is not necessary in Theorem 1 for the …rst inequality (1.4) to hold. A proof of this fact is proved below after some short preliminaries. The second inequality in (1.4) has been proved in [54] without differentiability assumption. 2. Preliminaries We recall some facts on the lateral derivatives of a convex function. Suppose that I is an interval of real numbers with interior I and f : I ! R is a convex function on I. Then f is continuous on I and has …nite left and right derivatives at each point of I. Moreover, if x; y 2 I and x < y; then f 0 (x) f 0 + (x) f 0 (y) f 0 + (y) which shows that both f 0 and f 0 + are nondecreasing function on I. It is also known that a convex function must be di¤erentiable except for at most countably many points. For a convex function f : I ! R, the subdi¤erential of f denoted by @f is the set of all functions ' : I ! [ 1;1] such that ' °I R and f (x) f (a) + (x a)' (a) for any x; a 2 I: It is also well known that if f is convex on I; then @f is nonempty, f 0 , f 0 + 2 @f and if ' 2 @f , then f 0 (x) ' (x) f 0 + (x) for any x 2 I. In particular, ' is a nondecreasing function. If f is di¤erentiable and convex on I, then @f = ff 0g : Now, since f exp is convex on [ln a; ln b] it follows that f has …nite lateral derivatives on (ln a; ln b) and by gradient inequality for convex functions we have (2.1) f exp (x) f exp (y) (x y)' (exp y) exp y where ' (exp y) 2 f 0 (exp y) ; f 0 + (exp y) for any x; y 2 (ln a; ln b) : INEQUALITIES OF HERMITE-HADAMARD TYPE FOR GA-CONVEX FUNCTIONS 3 If s; t 2 (a; b) and we take in (2.1) x = ln t; y = ln s; then we get (2.2) f (t) f (s) (ln t ln s)' (s) s where ' (s) 2 f 0 (s) ; f 0 + (s) : Now, if we take the integral mean on [a; b] in the inequality (2.2) we get 1 b a Z b a f (t) dt f (s) 1 b a Z b a ln tdt ln s ! ' (s) s and since 1 b a Z b a ln tdt = ln I (a; b) then we get (2.3) 1 b a Z b a f (t) dt f (s) + (ln I (a; b) ln s)' (s) s for any s 2 (a; b) and ' (s) 2 f 0 (s) ; f 0 + (s) : This is an inequality of interest in itself. Now, if we take in (2.3) s = I (a; b) 2 (a; b) then we get the …rst inequality in (1.4) for GA-convex functions. If f is di¤erentiable and GA-convex on (a; b) ; then we have from (2.3) the inequality (2.4) 1 b a Z b a f (t) dt f (s) + (ln I (a; b) ln s) f 0 (s) s for any s 2 (a; b) : If we take in (2.4) s = a+b 2 = A (a; b) ; then we get (2.5) 1 b a Z b a f (t) dt f (A (a; b)) f 0 (A (a; b))A (a; b) ln A (a; b) I (a; b) : If we assume that f 0 (A (a; b)) 0; then, since I (a; b) A (a; b) ; we get (2.6) 1 b a Z b a f (t) dt f (A (a; b)) provided that f is di¤erentiable and GA-convex on (a; b) : Also, if we take in (2.4) s = L (a; b) ; then we get (2.7) 1 b a Z b a f (t) dt f (L (a; b)) + f 0 (L (a; b))L (a; b) ln I (a; b) L (a; b) : If we assume that f 0 (L (a; b)) 0; then we get from (2.7) that (2.8) 1 b a Z b a f (t) dt f (L (a; b)) provided that f is di¤erentiable and GA-convex on (a; b) : Now, if we take in (2.4) s = p ab = G (a; b) ; then we get (2.9) 1 b a Z b a f (t) dt f (G (a; b)) + f 0 (G (a; b))G (a; b) ln I (a; b) G (a; b) : 4 S. S. DRAGOMIR Since ln I (a; b) G (a; b) = ln I (a; b) lnG (a; b) = b ln b a ln a b a 1 ln a+ ln b 2 = a+ b 2 ln b ln a b a 1 = A (a; b) L (a; b) L (a; b) ; then (2.9) is equivalent to (2.10) 1 b a Z b a f (t) dt f (G (a; b)) + f 0 (G (a; b))G (a; b) A (a; b) L (a; b) L (a; b) : If f 0 (G (a; b)) 0; then we have (2.11) 1 b a Z b a f (t) dt f (G (a; b)) provided that f is di¤erentiable and GA-convex on (a; b) : Motivated by the above results we establish in this paper other inequalities of Hermite-Hadamard type for GA-convex functions. Applications for special means are also provided. 3. New Results We start with the following result that provide in the right side of (1.4) a bound in terms of the identric mean. Theorem 2. Let f : [a; b] (0;1) ! R be a GA-convex (concave) function on [a; b] : Then we have 1 b a Z b a f (t) dt ( ) (ln b ln I (a; b)) f (a) + (ln I (a; b) ln a) f (b) ln b ln a (3.1) = b L (a; b) b a f (b) + L (a; b) a b a f (a) : Proof. Since is a GA-convex (concave) function on [a; b] then f exp is convex (concave) and we have f (t) = f exp (ln t) = f exp (ln b ln t) ln a+ (ln t ln a) ln b ln b ln a (3.2) ( ) (ln b ln t) f exp (ln a) + (ln t ln a) f exp (ln b) ln b ln a = (ln b ln t) f (a) + (ln t ln a) f (b) ln b ln a for any t 2 [a; b] : This inequality is of interest in itself as well. If we take the integral mean in (3.2) we get 1 b a Z b

Abstract: The bandwidth, average bandwidth, envelope, prole and antibandwidth of the matrices have been the subjects of study for at least 45 years. These problems have generated considerable interest over the years because of them practical relevance in ar- eas like: solving the system of equations, nite element methods, circuit design, hypertext layout, chemical kinetics, numerical geo- physics etc. In this paper a brief description of these problems are made in terms of their denitions, followed by a comparative study of them, using both approaches: matrix geometry and graph theory. Time evolution of the corresponding algorithms as well as a short description of them are made. The work also contains concrete real applications for which a large part of presented al- gorithms were developed.

Abstract: Let G =( V, E) be a simple connected molecular graph. In such a simple molecular graph, vertices represent atoms and edges represent chemical bonds, we denoted the sets of vertices and edges by V = V (G )a ndE = E(G), respectively. If d(u, v )b e the notation of distance between vertices u, v ∈ V and is defined as the length of a shortest path connecting them. Then, Eccen- tricity connectivity polynomial of a molecular graph G is defined as ECP(G, x )= v∈V dG(v)x ecc(v) ,w hereecc(v) is defined as the length of a maximal path connecting to another vertex of v. dG(v )( or simplydv) is degree of a vertex v ∈ V (G), and is defined as the number of adjacent vertices with v. In this paper, we fo- cus on the structure of molecular graph circumcoronene series of benzenoid Hk (k ≥ 2) and counting the eccentricity connectivity polynomial ECP(Hk) and eccentricity connectivity index ξ(Hk), by new method (called Ring-cut Method).