FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL FOR POWER SERIES IN BANACH ALGEBRAS VIA GRÜSS-LUPAŞ TYPE INEQUALITIES FOR p-norms

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1 Mem. Gra. Sci. Eng. Shimane Univ. Series B: Mathematics 49 06), pp FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL FOR POWER SERIES IN BANACH ALGEBRAS VIA GRÜSS-LUPAŞ TYPE INEQUALITIES FOR p-norms SILVESTRU SEVER DRAGOMIR, MARIUS VALENTIN BOLDEA, AND MIHAIL MEGAN Communicated by Takeshi Wada Received: August 5, 05) Abstract. Some Grüss-Lupaş type ineualities for p-norms of seuences in Banach algebras are obtained. Moreover, if f λ) = α nλ n is a function defined by power series with complex coefficients and convergent on the open disk D 0, R) C, R > 0 and x, y B, a Banach algebra, with xy = yx, then we also establish some new upper bounds for the norm of the Čebyšev type difference f λ) f λxy) f λx) f λy), λ D 0, R). These results build upon the earlier results obtained by the authors. Applications for some fundamental functions such as the exponential function and the resolvent function are provided as well.. Introduction In 935, G. Grüss [] proved the following integral ineuality which gives an approximation of the integral mean of the product in terms of the product of the integral means integrals as follows: ) b a b a f x) g x) dx b a b a f x) dx b a b a g x) dx Φ φ) Γ γ) 4 where f, g : [a, b] R are integrable on [a, b] and satisfying the assumption φ f x) Φ, γ g x) Γ for each x [a, b] where φ, Φ, γ, Γ are given real constants. Moreover the constant is sharp in the sense that it can not be replaced by a 4 smaller one. 000 Mathematics Subject Classification. 47A63; 47A99. Key words and phrases. Banach algebras, Power series, Exponential function, Resolvent function, Norm ineualities. 5

2 6 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN For a simple proof of ) as well as for some other integral ineualities of Grüss type see the Chapter X of the recent book [] by Mitrinović, Pe carić and Fink. In 950, M. Biernacki, H. Pidek and C. Ryll-Nardzewski [] established the following discrete version of Grüss ineuality, see also [, Ch. X]: Theorem.. Let a = a,..., a n ), b = b,..., b n ) be two n-tuples of real numbers such that r a i R and s b i S for i =,..., n. Then one has the ineuality: a i b i a i ) b i n n n [ n ] [ n ] ) R r) S s) n n when [x] is the integer part of x, x R. In 98, A. Lupaş [, Ch. X] proved some similar results for the first difference of a as follows : Theorem.. Let a, b two monotonic n-tuples in the same sense and p a positive n-tuple. Then ) min a i+ a i min b i+ b i i 3) p i ip i in in P n P n P n p i a i b i P n p i a i P n p i b i max a i+ a i max b i+ b i in in P n i p i P n ) ip i. If there exists the numbers ā, ā, r, r, rr > 0) such that a k = ā + kr and b k = ā + kr, then in 3) the euality holds. For some generalizations of Gruss ineuality for isotonic linear functionals defined on certain spaces of mappings see Chapter X of the book [] where further references are given. In order to extend the above results for Banach algebras, we need some preliminary facts as follows: Let B be an algebra. An algebra norm on B is a map : B [0, ) such that B, ) is a normed space, and, further: ab a b for any a, b B. The normed algebra B, ) is a Banach algebra if is a complete norm. We assume that the Banach algebra is unital, this means that B has an identity and that =. Let B be a unital algebra. An element a B is invertible if there exists an element b B with ab = ba =. The element b is uniue; it is called the inverse

3 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL 7 of a and written a or. The set of invertible elements of B is denoted by InvB. a If a, b InvB then ab InvB and ab) = b a. For a unital Banach algebra we also have: i) If a B and lim n a n /n <, then a InvB; ii) {a B: a < } InvB; iii) InvB is an open subset of B; iv) The map InvB a a InvB is continuous. For simplicity, we denote λ, where λ C and is the identity of B, by λ. The resolvent set of a B is defined by ρ a) := {λ C : λ a InvB} ; the spectrum of a is σ a), the complement of ρ a) in C, and the resolvent function of a is R a : ρ a) InvB, R a λ) := λ a). For each λ, γ ρ a) we have the identity R a γ) R a λ) = λ γ) R a λ) R a γ). Let f be an analytic functions on the open disk D 0, R) given by the power series f λ) := α jλ j λ < R). If ν a) < R, then the series α ja j converges in the Banach algebra B because α j a j <, and we can define f a) to be its sum. Clearly f a) is well defined and there are many examples of important functions on a Banach algebra B that can be constructed in this way. For instance, the exponential map on B denoted exp and defined as exp a := j! aj for each a B. It is known that if x and y are commuting, i.e. xy = yx, then the exponential function satisfies the property exp x) exp y) = exp y) exp x) = exp x + y). Also, if x is invertible and a, b R with a < b then b a exp tx) dt = x [exp bx) exp ax)]. Moreover, if x and y are commuting and y x is invertible, then 0 exp s) x + sy) ds = y x) [exp y) exp x)]. Ineualities for functions of operators in Hilbert spaces may be found in the recent monographs [5], [6], [0] and the references therein. Let α n be nonzero complex numbers and let R :=. lim sup α n n Clearly 0 R, but we consider only the case 0 < R. Denote by:

4 8 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN D0, R) = consider the functions: and { {λ C : λ < R}, if R < C, if R =, λ fλ) : D0, R) C, fλ) := λ f A λ) : D0, R) C, f A λ) := α n λ n α n λ n. Let B be a unital Banach algebra and its unity. Denote by { {x B : x < R}, if R < B0, R) = B, if R =. We associate to f the map: x fx) : B0, R) B, fx) := α n x n. Obviously, f is correctly defined because the series α nx n is absolutely convergent, since α nx n α n x n. In addition, we assume that s := n α n <. Let s 0 := α n < and s := n α n <. With the above assumptions we have that [7]: Theorem.3. Let λ C such that max{ λ, λ } < R < and let x, y B with x, y and xy = yx. Then: i) We have f λ ) f λxy) f λx) f 4) λy) ψ min {x, y } f A λ ) where: 5) ψ := s 0 s s. ii) We also have f λ ) f λxy) f λx) f 6) λy) min {x, y } f A λ ) { f A λ ) [ λ f A λ ) + λ f A λ ) ] [ λ f A λ )] } /. For other similar results, see [7], [8] and [9] Motivated by the above results we establish in this paper other similar ineualities for the norm of the Čebyšev difference f λ ) f λxy) f λx) f λy)

5 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL 9 via some Grüss -Lupaş type ineuality for p-norms with p, where λ is a complex number and the vectors x, y belong to the Banach algebra B. Applications for some fundamental functions such as the exponential function and the resolvent function are provided as well.. A Discrete Ineuality of Grüss Type for -Norm The following ineuality of Grüss type holds. Theorem.. Let B be a Banach algebra over K =R,C), a i, b i B and α i K i =,..., n). Then we have the ineuality: 7) α i α i a i b i α i a i α i b i n n α i ) α i a i b i, where a i := a i+ a i i =,..., n ) and b i := b i+ b i i =,..., n ) are the usual forward differences. The constant is sharp in the sense that it cannot be replaced by a smaller constant. Proof. Let us start with the following identity in Banach algebras which can be proved by direct computation α i α i a i b i α i a i α i b i = α i α j a j a i ) b j b i ) As i < j, we can write i,j= = i<jn α i α j a j a i ) b j b i ). j j j j a j a i = a k+ a k ) = a k and b j b i = b l+ b l ) = b l. k=i k=i Using the generalized triangle ineuality, we have successively: j 8) α i α i a i b i α i a i α i b i = α i α j i<jn k=i j j α i α j a k b l i<jn i<jn k=i l=i j j α i α j a k b l =: A. k=i l=i l=i l=i a k j b l l=i

6 0 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN It is obvious for all i < j n, we have that and and then j n a k a k k=i k= j n b l b l l=i l= n n 9) A a k b l k= l= i<jn α i α j. Now, let us observe that [ α i α j = α i α j ] 0) α i α j i<jn i,j= i=j [ ] = α i α j α i j= = α i ) α i. Using 8)-0), we deduce the desired ineuality 7). To prove the sharpness of the constant, let us assume that 7) holds with a constant c > 0. That is, ) α i α i a i b i α i a i α i b i n n c α i ) α i a i b i for all a i, b i, α i i =,..., n) as above and n. Choose in 7) n = and compute α i α i a i b i α i a i α i b i = α i α j a i a j ) b i b j ) i,j= = α i α j a i a j ) b i b j ) i<j = α α a a ) b b ).

7 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL Also, α i α j a i b i = α α a a b b. i<j Substituting in ), we obtain α α a a b b c α α a a b b. If we assume that α, α > 0, a a, b b, then we obtain c, which proves the sharpness of the constant. Remark.. Let B be a Banach algebra over K =R,C), a i B and α i K i =,..., n). Then we have the ineuality: ) ) α i α i a i α i a i n α i ) α i a i ). The constant is best possible. The following corollary holds. Corollary.3. Under the above assumptions for a i, b i i =,..., n), we have the ineuality 3) a i b i a i b i ) n n a i b i, n n n n and the constant is sharp. In particular, we have a i n n ) a i ) n n a i ). 3. An Ineuality of Grüss Type for p-norm The following result that provides a version for the p-norm, p > of the forward difference also holds.

8 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN Theorem 3.. Let B be a Banach algebra over K =R,C), a i, b i B and α i K i =,..., n). Then we have the ineuality: 4) α i α i a i b i α i a i α i b i j<in i j) α i α j n ) ) p n a k p b k, k= where p >, p + =. The constant C = in the right hand side of 7) is sharp in the sense that it cannot be replaced by a smaller one. Proof. From the proof of Theorem. we have 5) α i α i a i b i α i a i j<in α i b i k= i i α i α j a k b l =: A. k=j Using Hölder s discrete ineuality, we can state that and i a k i j) k=j i b k i j) p k=j where p, > and p + 6) A As and j<in i k=j i k=j l=j a k p ) p b k ) =, and then, by multiplication, we have α i α j i j) i ) ) p i a k p b k. k=j i n a k p a k p k=j k= i n b k b k, k=j for all j < i n, then by 5) and 6), we get the desired ineuality 4). To prove the sharpness of the constant, let us assume that 4) holds with a constant C > 0. That is, k= k=j

9 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL 3 7) C α i α i a i b i j<in α i a i i j) α i α j α i b i n k= ) p n a k p k= b k ). Choose n =. Then α i α i a i b i and j<i α i a i α i b i = α α a a b b ) ) p i j) α i α j a k p b k k= = α α a a b b. Therefore, from 7), we obtain k= α α a a b b C α α a a b b for all a a, b b, and then C, which proves the sharpness of the constant. Remark 3.. A coarser upper bound, which can be more useful may be obtained by applying Cauchy-Schwartz s ineuality: j<in i j) α i α j j<in α i α j ) j<in α i α j i j) ) and taking into account that α i α j = α i ) j<in α i and j<in α i α j i j) = α i α j i j) i,j= ) = α i i α i i α i.

10 4 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN Thus, from 4), we can state the ineuality 8) α i α i a i b i α i a i α i b i α i ) α i α i n k= k= ) ) p n a k p b k, where p >, p + =. The following corollary holds. Corollary 3.3. With the above assumptions, we have 9) n a i b i n n 6n n k= where p >, + =. p The constant is the best possible. 6 a i n b i k= ) i α i i α i ) ) p n a k p b k Proof. The proof follows by 7), putting α i = n and taking into account that j<in = j i j) j) + j3 3 j) jn n j) = + ) ) n n n) = n + ) + + 3) n) ) = k k k + ) = k k = n n ), 6 k= k= k= k= and the corollary is thus proved.

11 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL 5 Remark 3.4. If in 4) and 9) we assume that p = =, then we get the ineualities: 0) α i α i a i b i α i a i α i b i and ) j<in n i j) α i α j a i b i n n 6n n k= n ) ) n a k b k k= a i n b i k= k= ) ) n a k b k, respectively. We also have the ineuality ) α i α i a i b i α i a i α i b i α i ) α i α i n k= k= ) ) n a k b k. In the case when b i = a i, i {,..., n} we get from 0) ) 3) α i α i a i α i a i i j) α i α j j<in and from ) ) 4) α i α i a i α i a i α i ) α i α i n ) a k. k= ) i α i i α i n ) a k k= ) i α i i α i

12 6 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN if 4. Ineualities for Power Series As some natural examples that are useful for applications, we can point out that, 5) f λ) = g λ) = h λ) = l λ) = ) n n λn = ln, λ D 0, ) ; + λ n= ) n n)! λn = cos λ, λ C; ) n n + )! λn+ = sin λ, λ C; ) n λ n =, λ D 0, ) ; + λ then the corresponding functions constructed by the use of the absolute values of the coefficients are 6) f A λ) = g A λ) = h A λ) = l A λ) = n= n λn = ln, λ D 0, ) ; λ n)! λn = cosh λ, λ C; n + )! λn+ = sinh λ, λ C; λ n =, λ D 0, ). λ

13 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL 7 Other important examples of functions as power series representations with nonnegative coefficients are: 7) exp λ) = n! λn λ C, ) + λ ln = λ n λn, λ D 0, ) ; n= sin Γ ) n + λ) = λ n+, λ D 0, ) ; π n + ) n! tanh λ) = F α, β, γ, λ) = n= λ D 0, ) ; where Γ is Gamma function. The following new result holds: n λn, λ D 0, ) Γ n + α) Γ n + β) Γ γ) n!γ α) Γ β) Γ n + γ) λn, α, β, γ > 0, Theorem 4.. Let fλ) = α nλ n be a power series that is convergent on the open disk D0, R), with R > 0. If x, y B with xy = yx and x, y <, then we have for λ C with λ < R the ineuality: f λ ) f λxy) f λx) f 8) λy) where x y [ f x) y) A λ ) f A λ )], 9) f A λ) := has the radius of convergence R. α n λ n Proof. From the ineuality 7) we have 30) α i λ i α i λ i xy) i α i λ i x i α i λ i y i ) n α i λ i α i λ i x j+ x j n y j+ y j, ) = n α i λ i α i λ i x j x ) n y j y )

14 8 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN ) x y α i λ i ) = x y α i λ i xn x y n y n α i λ i α i λ i x j n y j for any n. Since all the series whose partial sums are involved in 30) are convergent, then by letting n in 30) we deduce the desired ineuality 8). Corollary 4.. Let fλ) = α nλ n be a power series that is convergent on the open disk D0, R), with R > 0. If x B and x <, then we have for λ C with λ < R the ineuality: 3) f λ ) f λx ) [ ] f λx) x [ f x) A λ ) f A λ )]. We have the following result as well: Theorem 4.3. Let fλ) = α nλ n be a power series that is convergent on the open disk D0, R), with R > 0. If x, y B with xy = yx and x, y <, then we have for λ C with λ < R the ineuality: 3) f λ ) f λxy) f λx) f λy) x y [ f x p ) p y ) A λ ) f A λ )] / { f A λ ) [ λ f A λ ) + λ f A λ ) ] [ λ f A λ )] } / where p >, p + =. In particular, we have 33) f λ ) f λx ) [ ] f λx) x [ f x p ) p x ) A λ ) f A λ )] / { f A λ ) [ λ f A λ ) + λ f A λ ) ] [ λ f A λ )] } /.

15 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL 9 Proof. Using the ineuality 8) we have for n that 34) α i λ i α i λ i xy) i α i λ i x i α i λ i y i ) / α i λ i α i λ i ) α i λ i i α i λ i i α i λ i n ) x j+ x j p p n ) y j+ y j, where p >, p + =. Observe that n x j+ x j p n x j x ) p n x p x j p n x p x jp = x p x np x p, which implies that n ) x j+ x j p p x np x x p ) p. Similarly, n ) y j+ y j y n y y ), where p >, p + =.

16 30 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN From 34) we get 35) α i λ i α i λ i xy) i α i λ i x i α i λ i y i ) / α i λ i α i λ i ) α i λ i i α i λ i i α i λ i x np x y x p ) p y n y where p >, p + =. If we denote g u) := α nu n, then for u < R we have and However and then Therefore and nα n u n = ug u) n α n u n = u ug u)). u ug u)) = ug u) + u g u) n α n u n = ug u) + u g u). n α n λ n = λ f A λ ) + λ f A λ ) n α n λ n = λ f A λ ) ), for λ < R. Since all the series whose partial sums are involved in 35) are convergent, then by letting n in 35) we deduce the desired ineuality 3).

17 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL 3 Corollary 4.4. With the assumptions of Theorem 4.3, we have f λ ) f λxy) f λx) f 36) λy) x y [ x ) y ) f A λ ) f A λ )] / { f A λ ) [ λ f A λ ) + λ f A λ ) ] [ λ f A λ )] } / and, in particular f λ ) f λx ) [ ] 37) f λx) x [ f x A λ ) f A λ )] / { f A λ ) [ λ f A λ ) + λ f A λ ) ] [ λ f A λ )] } /. 5. Some Particular Cases of Interest Consider the function f : D 0, ) C defined by Then which implies that f λ) = λ) = f A λ) := λ k = f A λ). k=0 λ n = λ), fa λ ) f A λ ) = λ λ ) + λ ), λ < and by 8), we have for x, y B with xy = yx, x, y < and λ C with λ < that 38) λ) λxy) λx) λy) 39) We also have for λ, x < that λ x y λ ) + λ ) x) y). λ) λx ) λx) λ x λ ) + λ ) x).

18 3 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN For the function f λ) = λ) we have f A λ ) [ λ f A λ ) + λ f A λ ) ] [ λ f A λ )] [ ] λ = λ λ ) + λ λ λ ) 3 λ ) 4 = λ λ ) 4. From the ineuality 3) we then have for x, y B with xy = yx, x, y < and λ C with λ < that λ) λxy) λx) λy) [ x y x p ) p y ) { } / λ λ ) 4, λ λ ) + λ ), which is euivalent to 40) λ) λxy) λx) λy) where p >, p + =. If we consider the function λ x y x p ) p y ) λ ) 3 + λ ) /, f λ) = + λ) = ) k λ k, then the ineualities 38)-40) also holds with + instead of in the left hand side expressions such as λx) etc. We consider the modified Bessel function functions of the first kind ) ν I ν λ) := λ λ) k 4 k!γ ν + k + ), λ C k=0 where Γ is the Gamma function and ν is a real number. An integral formula to represent I ν is I ν λ) = π π 0 e λ cos θ cos νθ) dθ which simplifies for ν an integer n to I n λ) = π π 0 k=0 sin νπ) π e λ cos θ cos nθ) dθ. 0 ] / e λ cosh t νt dt,

19 FURTHER BOUNDS FOR ČEBYŠEV FUNCTIONAL 33 For n = 0 we have I 0 λ) = π π 0 e λ cos θ dθ = λ) k 4 k!), λ C. k=0 Now, if we consider the exponential function then for ρ > 0 we have f λ) = exp λ) = f A ρ) = k=0 k=0 k! λk, k!) ρk = I 0 ρ), which implies that fa λ ) f A λ ) = exp λ ) I 0 λ ), λ C. Making use of the ineuality 8), we have for x, y B with xy = yx, x, y < and λ C that 4) exp λ xy + )) exp λ x + y)) x y x) y) [exp λ ) I 0 λ )], In particular, we have for x < 4) exp λ x + )) exp λx) x x) [exp λ ) I 0 λ )] for any λ C. For f λ) = exp λ) we have f A λ ) [ λ f A λ ) + λ f A λ ) ] [ λ f A λ )] = λ exp λ ). If x, y B with xy = yx and x, y <, then from 3) we have for λ C the ineuality: 43) exp λ xy + )) exp λ x + y)) x y λ / exp λ ) [exp λ ) I x p ) p y ) 0 λ )] /, where p >, p + =. Acknowledgment. The authors would like to thank the anonymous referee for valuable comments that have been implemented in the final version of the paper.

20 34 S. S. DRAGOMIRR, M. V. BOLDEA, AND M. MEGAN References [] M. Biernacki, Sur une inégalité entre les intégrales due à Tchebyscheff. Ann. Univ. Mariae Curie-Sklodowska Poland), A595), 3-9. [] K. Boukerrioua, and A. Guezane-Lakoud, On generalization of Čebyšev type ineualities. J. Ineual. Pure Appl. Math ), no., Article 55, 4 pp. [3] P. L. Čebyšev, O približennyh vyraženijah odnih integralov čerez drugie. Soobšćenija i protokoly zasedaniĭ Matemmatičeskogo občestva pri Imperatorskom Har kovskom Universitete No., 93 98; Polnoe sobranie sočineniĭ P. L. Čebyševa. Moskva Leningrad, 948a, 88), 8-3. [4] P. L. Čebyšev, Ob odnom rjade, dostavljajušćem predel nye veličiny integralov pri razloženii podintegral noĭ funkcii na množeteli. Priloženi k 57 tomu Zapisok Imp. Akad. Nauk, No. 4; Polnoe sobranie sočineniĭ P. L. Čebyševa. Moskva Leningrad, 948b, 883), [5] S. S. Dragomir, Operator Ineualities of the Jensen, Čebyšev and Grüss Type. Springer Briefs in Mathematics. Springer, New York, 0. xii+ pp. ISBN: [6] S. S. Dragomir, Operator Ineualities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, 0. x+ pp. ISBN: [7] S. S. Dragomir, M. V. Boldea, C. Buşe and M. Megan, Norm ineualities of Čebyšev type for power series in Banach algebras, Journal of Ineualities and Applications 04, 04:94 Online [8] S. S. Dragomir, M. V. Boldea and M. Megan, New norm ineualities of Čebyšev type for power series in Banach algebras, Preprint RGMIA Res. Rep. Coll., 7 04), Art. 65. Online [9] S. S. Dragomir, M. V. Boldea and M. Megan, New bounds for Čebyšev functional for power series in Banach algebras via a Grüss -Lupaş type ineuality, Preprint RGMIA Res. Rep. Coll., 7 04), Art. 04. Online [0] T. Furuta, J. Mićić Hot, J. Pečarić and Y. Seo, Mond-Pečarić Method in Operator Ineualities. Ineualities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, b [] G. Grüss, Über das Maximum des absoluten Betrages von a fx)gx)dx b b a) a fx)dx b a gx)dx, Math. Z., 39935), 5-6. [] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Ineualities in Analysis, Kluwer Academic Publishers, Dordrecht, 993. [3] D. S. Mitrinović and P. M. Vasić, History, variations and generalisations of the Čebyšev ineuality and the uestion of some priorities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No ), 30. Silvestru Sever Dragomir: Mathematics, School of Engineering & Science, Victoria University, PO Box 448, Melbourne City, MC 800, Australia address: sever.dragomir@vu.edu.au Marius Valentin Boldea: Mathematics and Statistics, Banat University of Agricultural Sciences and Veterinary Medicine Timişoara, 9 Calea Aradului, Timişoara, România address: marius.boldea@usab-tm.ro Mihail Megan: Department of Mathematics, West University of Timişoara, B- dul V. Pârvan 4, 900-Timişoara, România address: megan@math.uvt.ro b a