Some New Inequalities for a Sum of Exponential Functions

Transcription

1 Applied Mathematical Sciences, Vol. 9, 2015, no. 109, HIKARI Ltd, Some New Inequalities for a Sum of Eponential Functions Steven G. From Department of Mathematics University of Nebraska at Omaha Omaha, Nebraska USA Copyright c 2015 Steven G. From. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we present some new inequalities for sums of eponential functions which improve upon upper and lower estimates given in Ahmad [2004]. In some cases, a more general choice of eponential function parameters is allowed. Numerical comparisons are also made. Besides upper and lower bounds for minimum values, we present a very accurate approimation to the minimum values considered in Ahmad [2004] which is applicable to functions belonging to the absolutely monotonic class. Keywords: Absolutely monotonic; delay differential equation; eponential functions: inequalities 1 Introduction In Ahmad [2004], the result min >0 e T = T e (1) was generalized in the form of Theorems A and B below, for T > 0:

2 5430 Steven G. From Theorem A (Theorem 2.1 of Ahmad [2004].) Let p > 0, q > 0, and T > 0. Then pt ep 1 + q q pe e pe pe 2 T ( + q min pt ep 1 + q ). (2) >0 pe Theorem B (Theorem 2.2 of Ahmad [2004].) Let p i > 0, T i > 0, i = 1,..., n; T = ma 1 i n T i. Then L A e p i T i min >0 p i e T i ep 1 + p i (T T i ) U A. (3) e p i T i Theorems A and B were motivated by considering characteristic equations of various delay differential equations, which often involve sums of eponential functions divided by when considering oscillation conditions for these delay differential equations. For eamples of such equations and applications, see Ahmad [2004], p Main Results In this paper, we propose new upper and lower estimates/bounds and compare them both analytically and numerically to those given by (2) and (3) in Theorems A and B above. We also propose very accurate approimations to the minimum values in Theorems A and B below when bounds are not required, just very accurate approimations. We shall allow T i = 0 in Theorem B. First we give some new upper bounds for M 1 = min >0 pe T + q and compare them to the upper bound in Theorem A above. Theorem 2.1 below gives a new upper bound for M 1 and compares it to the upper bound in Theorem A. Part c) of Theorem 2.1 says U 1 below is an improvement of the upper bound in Theorem A above. Theorem 2.1 (4) a) M 1 (pe + q)t U 1

3 Some new inequalities 5431 ( b) M 1 pt pe p+q p p+q c) U 1 pt ep ( 1 + q pe). ) + q U1 Proof. Let f() = pet +q, > 0. Differentiation of f gives Also, f () = 2 [pe T (T 1) q] < 2 (T 1)pe T. (5) f () = 3 [ pe T ((T 1) 2 + 1) + 2q ] > 0 for > 0. So f is conve for > 0 and f is strictly increasing for > 0. Since f ( + ) as 0 + and f is conve, the equation f () = 0 has a unique root, call it 0, satisfying 0 T 1 0, by (5) above. Also, ( ) 1 f f( 0 ) = M 1. T But f ( ) 1 T = U1. Thus, M 1 U 1 and part a) is proven. To prove b), we modify slightly the proof of part (a). Then f () > 2 [ e T (pt (p + q)) ] = 0 when = p+q. Using the same argument as above, pt ( ) p + q U 2 = f f( 0 ) = M 1. pt This proves b). To prove c), let t = q. Then for t 0, pe e1+t e(1 + t) = e + q, from which p we obtain ( ) ( ) pt ep 1 + q pe pt e + q p = (pe + q)t = U 1. This completes the proof of (c) and the proof of the theorem is complete. We shall give a new lower bound/estimate for M 1 as a special case of a more general result to be given net. This new lower bound will improve on both lower bounds given in Theorems A and B above. First, to present more bounds, we need to discuss some results on absolutely monotonic functions.

4 5432 Steven G. From Definition 2.2 A function g() is absolutely monotonic on (0, ) if g() has the form g() = where H(t) is bounded and nondecreasing on (0, ). 0 e t dh(t), (6) Remark 2.3 In Sinik [2014], it was noted that the main theorem for absolutely monotonic functions on (0, ) from the book by Mitronovic, et al. [1993] was incorrect. Originally, an absolutely monotonic function g() on (0, ) was defined to have derivatives of all orders on (0, ) with q (k) () 0, 0 < <. (7) But Definitions (6) and (7) are not equivalent, it turns out. However, we are interested in Definition (6) above, since the numerators of the function f() to be minimized have form (6) above. For this definition, it is well-known that for > 0, g (k) ()g (k+2) () (g (k+1) ()) 2, k = 0, 1, 2,.... (8) See, for eample, Mitrinovic, Pecaric and Fink [1993], p See also Widder [1946], p Let s now consider upper and lower bounds for the minimum value M 2 = min f(), (9) >0 where f() = g() and g() = p i e Ti, (10) Clearly, g() absolutely monotonic on (0, ) and has form (6). We now present several new upper and lower bounds for M 2 and compare them to the bounds of Ahmad [2004] given in Theorem B, both analytically and numerically. We also allow some, but not all, of the T i values to be zero. Theorem 2.4 below presents new upper and lower bounds for M 2. The lower bound of Theorem B is embedded in a family of lower bounds for M 2, all at least as large as the lower bound of Theorem B. Theorem 2.4 Suppose p i > 0, i = 1, 2,..., n and T i 0, not all zero, i = 1, 2,..., n. Let p i T i T ave = and p i T ma = ma{t i : 1 i n}. Let = 1 T ma, = 1 T ave. Let 0 c. Then we have the following:

5 Some new inequalities 5433 (a) Let h 1 (c) = g(c) = n p i e Tic, h 2 (c) = g (c) = n p i T i e Tic. Let A(c) = h 2(c). Then M h 1 (c) 2 h 2 (c) e 1 c A(c) L(c). In addition, L(c) is nondecreasing in c, 0 c. Thus, the best possible lower bound is L( ). (b) Let 0 c. Let D = g ( ) g( ) = Then M 2 D h 1 (c)e 1 cd U(c). p i T i e T i. p i e T i (c) L(c) L(0) = ( n p i T i ) e = L A, the lower bound of Theorem B. Thus, L(c) is at least as good as the lower bound of Ahmad [2004]. (d) U(c) is nonincreasing in c, 0 c. Thus, the best such upper bound for M 2 is U( ). Proof. Let w() = Ln(g()) = Ln ( n p i e T i ). Then w() has derivatives w () = g () g() and w () = g()g () (g ()) 2 (g()) 2, > 0. Since g is absolutely monotonic on (0, ), w () 0, > 0. Differentiating f() = g(), we obtain ) f () = ( 2 p i T i e Ti p i e T i (11) ( n ) = 2 p i (T i 1)e T i. (12) Now if T i = 0, T i 1 = 1 < 0. If <, T i 1 < 0 also. Thus, 0 < gives f () < 0. From (11), we obtain ( n ) f () = 2 p i Q i (T i 1)e Ti, (13) p i where Q i = n, i = 1, 2,..., n. Note that Q k=1 p i 0, i = 1, 2,..., n and k n Q i = 1. Let J (T ) = (T 1)e T, > 0, T > 0. Then differentiating with respect to T, we obtain J (T ) = T 2 e T and J (T ) = (T )e T 0.

6 5434 Steven G. From Thus, J (T ) is a conve function of T for each > 0. inequality, we obtain Applying Jensen s ( n ( n ) ( f () 2 n ) ) Q i T i ) p i ( Q i T i 1 e = 0, when = 1 T ave =. Thus, > implies f () > 0. Thus, a minimum f() must occur on the interval [, ]. The Mean Value Theorem or a first degree Taylor epansion gives w() = w(c) + w (θ) ( c), (14) where θ is a real number in (c, ). Since w is increasing, we obtain Thus, w() w(c) + w (c) ( c). (15) Eponentiation of both sides of (15) gives, using w (c) = A(c), g() [ h 1 (c) e ca(c)] e A(c). (16) [h 1 (c)e ca(c) ] c A(c) min f() min >0 >0 = h 1 (c)e c A(c) min >0 = h 1 (c)e ca(c) A(c) e = h 2 (c)e 1 ca(c) = L(c). e A(c) This proves part (a). To prove (b), we use similar arguments. Then (14) gives instead Eponentiating gives w() w(c) + w ( )( c) = w(c) + D( c). (17) min f() h 1(c) e cd e D min >0 >0 = Dh 1 (c)e 1 cd = u(c).

7 Some new inequalities 5435 To prove (c), consider (15). Then w() h (c) w(c) + w (c) ( c). Fiing in [, ] and differentiating the right-hand side with respect to c, we obtain h (c) = w (c) + w (c) ( 1) + ( c)w (c) = ( c)w (c) 0, since 0 c and gives c 0. Thus, the lower bound function h (c) considered as a function of c is nondecreasing in c. Thus, L(c) is nondecreasing in c for all in [, ]. In particular, L(c) L(0) = ( n p i T i ) e, which is the Ahmad lower bound for M 2 given in Theorem B. From (17), we have w() w(c) + D( c) H (c). Differentiating H (c) with respect to c gives, for any in [, ], H (c) = w (c) D = w (c) w ( ) 0, since c and w 0 on (0, ). Thus H (c) is nonincreasing in c for each with [, ] which gives, upon eponentiation, that U(c) is nonincreasing on [0, ]. This proves (d). Remark 2.5 It is conjectured that U( ) U A. Not one case where U( ) U A has been found in many numerical comparisons done, some of which is given in Section 3 later. In any case, Theorem 3 presented net gives a new upper bound, U 2, which satisfies U 2 U A always. (It was noticed that U( ) U 2 as well, but this could not be proven either.) Thus, L( ) and U 2 improve upon L A and U A, respectively. Net, we present another upper bound of M 2 which sometimes is better than U(c) and which is at least as good as U A, the Ahmad upper bound of Theorem B. Theorem 2.6 We have M 2 (pe + q) U 2, where Also, U 2 U A. p = p i T i and q = p i (T ma T i ).

8 5436 Steven G. From Proof. In Ahmad [2004], it is established that M 2 min y>0 pe y + q y. (18) By Theorem 1, part (a) with the above choices of p and q and T = 1, the result M 2 U 2 follows immediately. The second part follows immediately from Theorem 1, part (c) with the above choices of p, q and T, since U A is also based on an application of (18) above. See Ahmad [2004], p. 4. Remark 2.7 We have established that L A L(c) and U 2 U A in Theorems 2 and 3, improving on both bounds of Theorem B. The best lower bound L(c) value occurs when c =. Net, we present a very accurate approimation of M 2, but necessarily bounds for M 2. The bounds L(c) and U(c) were found by taking natural logarithms of g() = n p i e Ti, that is w() = Ln(g()) bounds were found. Here, we use instead w() = g() δ for a suitable δ < 0. How to choose δ? One way is so that w() = (g()) δ is very nearly linear. This requires w () = 0. After some algebra, this produces the choice δ satisfying Solving (19) for δ, we obtain g()g () + (g ()) 2 (δ 1) = 0. (19) δ = 1 g()g () (g ()) 2 0. (20) (If δ = 0, a L Hospital s Rule type of argument applied here leads to w() = Ln(g()). It can be shown δ = 0 if and only if g() = ae b.) We shall use (20) for =, since L( ) is usually substantially closer to M 2 than is U( ). This will be seen later in numerical comparisons. Thus, we consider the choice Then we obtain: the approimation (not bounds) δ = δ = 1 g( )g ( ) (g ( )) 2. (21) w() [g( ) δ δ (g( )) δ 1 g ( )] + [δ(g( )) δ 1 g ( )] A + B which gives So g() (A + B ) 1 δ. g() M 2 = min f() = min >0 >0 min >0 (A + B ) 1 δ.

9 Some new inequalities 5437 The latter minimum value occurs at = A X B (. Corresponding mini- 1 δ 1) mum value ( A 1 δ) 1 δ. Thus, the approimation to M 2 is evaluated at δ = δ. M 2 = ( A 1 δ ) 1 δ, (22) 3 Numerical Comparisons In this section, we compare the bounds (L A and U A of Theorem B) of Ahmad (2004) to the new bounds (L( ), U( ) and U 2 ) presented in this paper. We present the bounds for p i e T i M 2 = min >0 for n = 2 first. This choice of n was used by Ahmad (2004) also. The first three rows of Table 1 below correspond to choices for p 1, p 2, T 1, and T 2 used by Ahmad (2004) in his Section 3 table. We also present the value of the approimation M2 (but not necessarily a bound) discussed in Section 2. Table 1 (Lower Bounds for M 2 and M 2 ) p 1 p 2 T 1 T 2 L A L( ) M 2 M Table 2 (Upper Bounds for M 2 ) p 1 p 2 T 1 T 2 M 2 U A U( ) U

10 5438 Steven G. From From Tables 1 and 2, we see that the new lower and upper bounds greatly improve ont he bounds of Ahmad (2004) when T 1 and T 2 differ greatly. Only when T 1 and T 2 are nearly equal are the new bounds only slightly better. We see that the approimation M 2 is a very good one also in most cases. For n 3, we present a few cases in Tables 3 and 4 below. Case A: n = 3, p 1 = 2, T 1 = 10, p 2 = 1, T 2 = 1, p 3 = 5, T 3 = 0.10 Case B: n = 3, p 1 = 1, T 1 = 5, p 2 = 5, T 2 = 0.2, p 3 = 4, T 3 = 20.0 Case C: n = 3, p 1 = 10, T 1 = 2, p 2 = 3, T 2 = 3, p 3 = 0.10, T 3 = 10.0 Table 3 (Lower Bounds for M 2 for Cases A, B, C and M 2 ) Case L A L( ) M 2 M 2 A B C Table 4 (Upper Bounds for M 2 for Cases A, B, C) Case M 2 U A U( ) U 2 A B C Again, we see that the new bounds are substantially better when the T i values are quite diverse. Also, M2 remains a very good approimation of M 2. These conclusions remain the same for n 4. The new bounds presented in this paper can be etended to bounds for f() = g(), where g() is absolutely monotonic on (0, ) having form (6).

11 Some new inequalities 5439 References [1] F. Ahmad, Inequalities for a sum of eponential functions, Journal of Inequalities in Pure and Applied Mathematics, 5 (2004), no. 4, 1 4. [2] D.S. Mitrinović, J. E. Pecarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, [3] S. M. Sitnik, On the correctness of the main theorem for absolutely monotonic functions, Research Group on Mathematics, Inequalities and Applications, (2014). [4] D. V. Widder, The Laplace Transform, Princeton University Press, Received: July 21, 2015; Published: August 25, 2015