The radius of α -convexity of normalized Bessel functions of the first kind
aa r X i v : . [ m a t h . C A ] D ec THE RADIUS OF α -CONVEXITY OF NORMALIZED BESSEL FUNCTIONS OFTHE FIRST KIND ´ARP ´AD BARICZ, HALIT ORHAN, AND R ´OBERT SZ ´ASZ Abstract.
The radii of α -convexity are deduced for three different kind of normalized Bessel functionsof the first kind and it is shown that these radii are between the radii of starlikeness and convexity, when α ∈ [0 , , and they are decreasing with respect to the parameter α. The results presented in this paperunify some recent results on the radii of starlikeness and convexity for normalized Bessel functions of thefirst kind. The key tools in the proofs are some interlacing properties of the zeros of some Dini functionsand the zeros of Bessel functions of the first kind. Introduction and the main results
Let D (0 , r ) be the open disk { z ∈ C : | z | < r } , where r > , and set D = D (0 , A we meanthe class of analytic functions f : D (0 , r ) → C which satisfy the usual normalization conditions f (0) = f ′ (0) − . Denote by S the class of functions belonging to A which are univalent in D (0 , r ) and let S ∗ ( β ) be the subclass of S consisting of functions which are starlike of order β in D (0 , r ) , where 0 ≤ β < . The analytic characterization of this class of functions is S ∗ ( β ) = (cid:26) f ∈ S : Re (cid:18) zf ′ ( z ) f ( z ) (cid:19) > β for all z ∈ D (0 , r ) (cid:27) , while the real number r ∗ β ( f ) = sup (cid:26) r > (cid:18) zf ′ ( z ) f ( z ) (cid:19) > β for all z ∈ D (0 , r ) (cid:27) is called the radius of starlikeness of order β of the function f. Note that r ∗ ( f ) = r ∗ ( f ) is the largestradius such that the image region f ( D (0 , r ∗ ( f ))) is a starlike domain with respect to the origin. Also, let K ( β ) be the subclass of S consisting of functions which are convex of order β in D (0 , r ) , where 0 ≤ β < . The well-known analytic characterization of this class of functions is K ( β ) = (cid:26) f ∈ S : Re (cid:18) zf ′′ ( z ) f ′ ( z ) (cid:19) > β for all z ∈ D (0 , r ) (cid:27) , and the real number r cβ ( f ) = sup (cid:26) r > (cid:18) zf ′′ ( z ) f ′ ( z ) (cid:19) > β for all z ∈ D (0 , r ) (cid:27) is called the radius of convexity of order β of the function f. Note that r c ( f ) = r c ( f ) is the largest radiussuch that the image region f ( D (0 , r c ( f ))) is a convex domain with respect to the origin. Furthermore,let M ( α, β ) be the subclass of S consisting of functions which are α − convex of order β in D (0 , r ) , where α ∈ R and 0 ≤ β < . The analytic characterization of this class of functions is M ( α, β ) = (cid:26) f ∈ S : Re (cid:18) (1 − α ) zf ′ ( z ) f ( z ) + α (cid:18) zf ′′ ( z ) f ′ ( z ) (cid:19)(cid:19) > β for all z ∈ D (0 , r ) (cid:27) , while the real number r α,β ( f ) = sup (cid:26) r > (cid:18) (1 − α ) zf ′ ( z ) f ( z ) + α (cid:18) zf ′′ ( z ) f ′ ( z ) (cid:19)(cid:19) > β for all z ∈ D (0 , r ) (cid:27) is called the radius of α − convexity of order β of the function f. The radius of α − convexity of order β isthe generalization of the radius of starlikeness of order β and of the radius of convexity of order β. We File: alpha.tex, printed: 2018-7-04, 8.22
Mathematics Subject Classification.
Key words and phrases.
Normalized Bessel functions of the first kind; convex functions; starlike functions; α -convexfunctions; radius of convexity; radius of starlikeness; radius of α -convexity; Dini function; minimum principle for harmonicfunctions; zeros of Bessel functions. α -CONVEXITY OF BESSEL FUNCTIONS have r ,β ( f ) = r ∗ β ( f ) and r ,β ( f ) = r cβ ( f ) . For more details on starlike, convex and α -convex functionswe refer to [10, 12, 13] and to the references therein.The Bessel function of the first kind of order ν is defined by [14, p. 217] J ν ( z ) = X n ≥ ( − n n !Γ( n + ν + 1) (cid:16) z (cid:17) n + ν , z ∈ C . In this paper we focus on the following normalized forms f ν ( z ) = (2 ν Γ( ν + 1) J ν ( z )) ν = z − ν ( ν + 1) z + . . . , ν = 0 ,g ν ( z ) = 2 ν Γ( ν + 1) z − ν J ν ( z ) = z − ν + 1) z + 132( ν + 1)( ν + 2) z − . . . ,h ν ( z ) = 2 ν Γ( ν + 1) z − ν J ν ( √ z ) = z − ν + 1) z + . . . , where ν > − . We note that in fact for z ∈ C \ { } we have f ν ( z ) = exp (cid:18) ν Log (2 ν Γ( ν + 1) J ν ( z )) (cid:19) , where Log represents the principal branch of the logarithm, and in this paper every multi-valued functionis taken with the principal branch. We also mention that the univalence, starlikeness and convexity ofBessel function of the first kind were studied extensively in several papers. We refer to [1, 2, 3, 4, 5, 6,7, 8, 9, 11, 15, 16] and to the references therein.In this paper we make a further contribution to the subject by showing the following new sharp resultscontained in Theorems 1, 2 and 3. The proofs of Theorems 1, 2 and 3 can be found in section 2. Theorem 1. If ν > , α ≥ and β ∈ [0 , , then the radius of α -convexity of order β of the function f ν is the smallest positive root of the equation α (cid:18) rJ ′′ ν ( r ) J ′ ν ( r ) (cid:19) + (cid:18) ν − α (cid:19) rJ ′ ν ( r ) J ν ( r ) = β. The radius of α -convexity satisfies r α,β ( f ν ) ≤ j ′ ν, < j ν, , where j ν, and j ′ ν, denote the first positivezeros of J ν and J ′ ν , respectively. Moreover, the function α r α,β ( f ν ) is strictly decreasing on [0 , ∞ ) andconsequently we have r cβ ( f ν ) < r α,β ( f ν ) < r ∗ β ( f ν ) for all α ∈ (0 , , β ∈ [0 , and ν > . Figure 1.
The graph of the function r J ( a, f ( r )) for a ∈ { , . , . , . , } on [0 , . . It is worth to mention that the cases α = 0 and α = 1 of the above Theorem were considered recentlyin [4, Theorem 1(a)] and [6, Theorem 1.1]. Our Theorem 1 is a common generalization of these results. A. BARICZ, H. ORHAN, R. SZ´ASZ/THE RADIUS OF α -CONVEXITY OF BESSEL FUNCTIONS 3 Figure 1 illustrates the fact that if α ∈ [0 , , then the radius of α -convexity of the function f ν is betweenits radii of convexity and starlikeness, that is, r cβ ( f ν ) < r α,β ( f ν ) < r ∗ β ( f ν ) for all α ∈ (0 , , β ∈ [0 ,
1) and ν > . We considered the particular cases when β = 0 . , ν = 1 and α ∈ { , . , . , . , } . Theorem 2. If ν > − , α ≥ and β ∈ [0 , , then the radius of α -convexity of order β of the function g ν is the smallest positive root of the equation α − rJ ν +1 ( r ) J ν ( r ) + αr rJ ν +2 ( r ) − J ν +1 ( r ) J ν ( r ) − rJ ν +1 ( r ) = β. The radius of α -convexity satisfies r α,β ( g ν ) ≤ α ν, < j ν, , where α ν, is the first positive zero of the Dinifunction z (1 − ν ) J ν ( z ) + zJ ′ ν ( z ) . Moreover, the function α r α,β ( g ν ) is strictly decreasing on [0 , ∞ ) and consequently we have r cβ ( g ν ) < r α,β ( g ν ) < r ∗ β ( g ν ) for all α ∈ (0 , , β ∈ [0 , and ν > − . Figure 2.
The graph of the function r J ( a, g . r )) for a ∈ { , . , . , . , } on [0 , . . We also note that the cases α = 0 and α = 1 of the above Theorem were considered recently in [4,Theorem 1(b)] and [6, Theorem 1.2]. Our Theorem 2 is a common generalization of these results. Figure2 illustrates the fact that when α ∈ [0 ,
1] the radius of α -convexity of the function g ν is between its radiiof convexity and starlikeness, that is, r cβ ( g ν ) < r α,β ( g ν ) < r ∗ β ( g ν ) for all α ∈ (0 , , β ∈ [0 ,
1) and ν > − . We considered the particular cases when β = 0 . , ν = 0 . α ∈ { , . , . , . , } . Theorem 3. If ν > − , α ≥ and β ∈ [0 , , then the radius of α -convexity of order β of the function h ν is the smallest positive root of the equation (1 − α ) − r · J ν +1 ( r ) J ν ( r ) ! + α r · r J ν +2 ( r ) − J ν +1 ( r )2 J ν ( r ) − r J ν +1 ( r ) ! = β. The radius of α -convexity satisfies r α,β ( h ν ) ≤ β ν, < j ν, , where α β, is the first positive zero of the Dinifunction z (2 − ν ) J ν ( z ) + zJ ′ ν ( z ) . Moreover, the function α r α,β ( h ν ) is strictly decreasing on [0 , ∞ ) and consequently we have r cβ ( h ν ) < r α,β ( h ν ) < r ∗ β ( h ν ) for all α ∈ (0 , , β ∈ [0 , and ν > − . Finally , we mention that the cases α = 0 and α = 1 of the above Theorem were also considered recentlyin [4, Theorem 1(c)] and [6, Theorem 1.3]. Our Theorem 3 is a common generalization of these results.Figure 3 illustrates the fact that for α ∈ [0 ,
1] the radius of α -convexity of the function h ν is between itsradii of convexity and starlikeness, that is, r cβ ( h ν ) < r α,β ( h ν ) < r ∗ β ( h ν ) for all α ∈ (0 , , β ∈ [0 ,
1) and ν > − . We considered the particular cases when β = 0 . , ν = − . α ∈ { , . , . , . , } . We would liketo take the opportunity to mention that [4, Theorem 1(c)] should be corrected as follows: if ν > − , then r ∗ ( h ν ) = z ν,β, , where z ν,β, is the smallest positive root of the equation z J ′ ν ( z )+(2 − β − ν ) J ν ( z ) = 0 . In [4, Theorem 1(c)] the above result was stated wrongly with z instead of z . Consequently, [4, Corollary1(c)] should be rewritten accordingly as follows: if ν > − , then the radius of starlikeness of the function h ν is z ν, , , which denotes the smallest positive root of the equation z J ′ ν ( z ) + (2 − ν ) J ν ( z ) = 0 . ´A. BARICZ, H. ORHAN, R. SZ´ASZ/THE RADIUS OF α -CONVEXITY OF BESSEL FUNCTIONS Figure 3.
The graph of the function r J ( a, h − . r )) for a ∈ { , . , . , . , } on [0 , . . Proof of the main results
In this section our aim is to present the proofs of the main theorems. For convenience in the sequelwe will use the following notation J ( α, u ( z )) = (1 − α ) zu ′ ( z ) u ( z ) + α (cid:18) zu ′′ ( z ) u ′ ( z ) (cid:19) . Proof of Theorem 1.
Without loss of generality we assume that α > . The case α = 0 was provedalready in [4]. By using the definition of the function f ν we have zf ′ ν ( z ) f ν ( z ) = 1 ν zJ ′ ν ( z ) J ν ( z ) , zf ′′ ν ( z ) f ′ ν ( z ) = 1 + zJ ′′ ν ( z ) J ′ ν ( z ) + (cid:18) ν − (cid:19) zJ ′ ν ( z ) J ν ( z ) . In view of the following infinite product representations [14, p. 235] J ν ( z ) = (cid:0) z (cid:1) ν Γ( ν + 1) Y n ≥ (cid:18) − z j ν,n (cid:19) , J ′ ν ( z ) = (cid:0) z (cid:1) ν − ν ) Y n ≥ (cid:18) − z j ′ ν,n (cid:19) , where j ν,n and j ′ ν,n are the n th positive roots of J ν and J ′ ν , respectively, logarithmic differentiation yields zJ ′ ν ( z ) J ν ( z ) = ν − X n ≥ z j ν,n − z , zJ ′′ ν ( z ) J ′ ν ( z ) = ν − X n ≥ z j ′ ν,n − z , which implies that J ( α, f ν ( z )) = (1 − α ) zf ′ ν ( z ) f ν ( z ) + α (cid:18) zf ′′ ν ( z ) f ′ ν ( z ) (cid:19) = α + (cid:18) ν − α (cid:19) zJ ′ ν ( z ) zJ ν ( z ) + α zJ ′′ ν ( z ) J ′ ν ( z )= 1 − (cid:18) ν − α (cid:19) X n ≥ z j ν,n − z − α X n ≥ z j ′ ν,n − z . On the other hand, we know (see [6, Lemma 2.1]) that if a > b > , z ∈ C and λ ≤ , then for all | z | < b we have(2.1) λ Re (cid:18) za − z (cid:19) − Re (cid:18) zb − z (cid:19) ≥ λ | z | a − | z | − | z | b − | z | . Note that in [6, Lemma 2.1] it was assumed that λ ∈ [0 , , however, following the proof of [6, Lemma 2.1]it is clear that we do not need the assumption λ ≥ . By using the inequality (2.1) for all z ∈ D (0 , j ′ ν, ) A. BARICZ, H. ORHAN, R. SZ´ASZ/THE RADIUS OF α -CONVEXITY OF BESSEL FUNCTIONS 5 we obtain the inequality1 α Re J ( α, f ν ( z )) ≥ α + (cid:18) − αν (cid:19) X n ≥ r j ν,n − r − X n ≥ r j ′ ν,n − r = 1 α J ( α, f ν ( r )) , where | z | = r. Here we used that the zeros j ν,n and j ′ ν,n interlace according to the inequalities [14, p. 235](2.2) ν ≤ j ′ ν, < j ν, < j ′ ν, < j ν, < j ′ ν, < . . .. Now, the above deduced inequality implies that for r ∈ (0 , j ′ ν, ) we have inf z ∈ D (0 ,r ) J ( α, f ν ( z )) = J ( α, f ν ( r )) . On the other hand, the function r J ( α, f ν ( r )) is strictly decreasing on (0 , j ′ ν, ) since ∂∂r J ( α, f ν ( r )) = − (cid:18) ν − α (cid:19) X n ≥ rj ν,n ( j ν,n − r ) − α X n ≥ rj ′ ν,n ( j ′ ν,n − r ) < α X n ≥ rj ν,n ( j ν,n − r ) − α X n ≥ rj ′ ν,n ( j ′ ν,n − r ) < ν > r ∈ (0 , j ′ ν, ) . Here we used again that the zeros j ν,n and j ′ ν,n interlace and for all n ∈ N ,ν > r < q j ν, j ′ ν, we have that j ν,n ( j ′ ν,n − r ) < j ′ ν,n ( j ν,n − r ) . We also have that lim r ց J ( α, f ν ( r )) = 1 > β and lim r ր j ′ ν, J ( α, f ν ( r )) = −∞ , which means that for z ∈ D (0 , r ) we have Re J ( α, f ν ( z )) > β if and only if r is the unique root of J ( α, f ν ( r )) = β, situated in(0 , j ′ ν, ) . Finally, by using again the interlacing inequalities (2.2) we obtain the inequality ∂∂α J ( α, f ν ( r )) = X n ≥ r j ν,n − r − X n ≥ r j ′ ν,n − r < , where ν > , α ≥ r ∈ (0 , j ′ ν, ) . This implies that the function α J ( α, f ν ( r )) is strictly decreasingon [0 , ∞ ) for all ν > r ∈ (0 , j ′ ν, ) fixed. Consequently, as a function of α the unique root of theequation J ( α, f ν ( r )) = β is strictly decreasing, where β ∈ [0 , , ν > r ∈ (0 , j ′ ν, ) are fixed. Thus,in the case when α ∈ (0 ,
1) the radius of α -convexity of the function f ν will be between the radius ofconvexity and the radius of starlikeness of the function f ν . This completes the proof. (cid:3)
Proof of Theorem 2.
Similarly, as in the proof of Theorem 1 we assume that α > . The case α = 0was proved already in [4]. We start with the following relations zg ′ ν ( z ) g ν ( z ) = 1 − ν + zJ ′ ν ( z ) J ν ( z ) , z g ′′ ν ( z ) g ′ ν ( z ) = ν ( ν − J ν ( z ) + 2(1 − ν ) zJ ′ ν ( z ) + z J ′′ ν ( z )(1 − ν ) J ν ( z ) + zJ ′ ν ( z ) . The recurrence formula [14, p. 222] zJ ′ ν ( z ) = νJ ν ( z ) − zJ ν +1 ( z ) and the fact that J ν is a particularsolution of the Bessel differential equation imply that z g ′′ ν ( z ) g ′ ν ( z ) = z zJ ν +2 ( z ) − J ν +1 ( z ) J ν ( z ) − zJ ν +1 ( z ) , and using [6, Lemma 2.4] it follows that1 + z g ′′ ν ( z ) g ′ ν ( z ) = 1 − X n ≥ z α ν,n − z , where α ν,n is the n th positive zero of the Dini function z (1 − ν ) J ν ( z ) + zJ ′ ν ( z ) . Thus, we have that J ( α, g ν ( z )) = (1 − α ) zg ′ ν ( z ) g ν ( z ) + α (cid:18) zg ′′ ν ( z ) g ′ ν ( z ) (cid:19) = (1 − α ) (cid:18) − ν + zJ ′ ν ( z ) J ν ( z ) (cid:19) + α (cid:18) z zJ ν +2 ( z ) − J ν +1 ( z ) J ν ( z ) − zJ ν +1 ( z ) (cid:19) = 1 + ( α − X n ≥ z j ν,n − z − α X n ≥ z α ν,n − z . ´A. BARICZ, H. ORHAN, R. SZ´ASZ/THE RADIUS OF α -CONVEXITY OF BESSEL FUNCTIONS Applying the inequality (2.1) we have that1 α Re J ( α, g ν ( z )) ≥ α + (cid:18) − α (cid:19) X n ≥ r j ν,n − r − X n ≥ r α ν,n − r = 1 α J ( α, g ν ( r )) , where | z | = r. Here we used tacitly that for all n ∈ { , , . . . } we have α ν,n ∈ ( j ν,n − , j ν,n ) , where j ν,n isthe n th positive zero of J ν . This follows immediately from Dixon’s theorem [17, p. 480], which says thatwhen ν > − a, b, c, d are constants such that ad = bc, then the positive zeros of z aJ ν ( z )+ bzJ ′ ν ( z )are interlaced with those of z cJ ν ( z ) + dzJ ′ ν ( z ) . Thus, if we choose a = 1 − ν, b = 1 , c = 1 and d = 0 , then the required assertion follows. Note also that the zeros α ν,n are all real when ν > − , see [17,p. 482], and thus the application of the inequality (2.1) is allowed. Thus, for r ∈ (0 , α ν, ) we getinf z ∈ D (0 ,r ) Re J ( α, g ν ( z )) = J ( α, g ν ( r )) , since according to the minimum principle of harmonic functionsthe infimum is taken on the boundary. On the other hand, the function r J ( α, g ν ( r )) is strictlydecreasing on (0 , α ν, ) since ∂∂r J ( α, g ν ( r )) = ( α − X n ≥ rj ν,n ( j ν,n − r ) − α X n ≥ rα ν,n ( α ν,n − r ) < α X n ≥ rj ν,n ( j ν,n − r ) − α X n ≥ rα ν,n ( α ν,n − r ) < ν > − r ∈ (0 , α ν, ) . Here we used again that the zeros j ν,n and α ν,n interlace and for all n ∈ N ,ν > − r < p j ν, α ν, we have that j ν,n ( α ν,n − r ) < α ν,n ( j ν,n − r ) . We also have that lim r ց J ( α, g ν ( r )) = 1 > β and lim r ր α ν, J ( α, g ν ( r )) = −∞ , which means that for z ∈ D (0 , r ) we have Re J ( α, g ν ( z )) > β if and only if r is the unique root of J ( α, g ν ( r )) = β, situated in(0 , α ν, ) . Finally, by using again the interlacing inequalities j ν,n − < α ν,n < j ν,n we obtain the inequality ∂∂α J ( α, g ν ( r )) = X n ≥ r j ν,n − r − X n ≥ r α ν,n − r < , where ν > − , α ≥ r ∈ (0 , α ν, ) . This implies that the function α J ( α, g ν ( r )) is strictly decreasingon [0 , ∞ ) for all ν > − r ∈ (0 , α ν, ) fixed. Consequently, as a function of α the unique root ofthe equation J ( α, g ν ( r )) = β is strictly decreasing, where β ∈ [0 , , ν > − r ∈ (0 , α ν, ) are fixed.Thus, for α ∈ (0 ,
1) the radius of α -convexity of the function g ν is between the radius of convexity andthe radius of starlikeness of the function g ν . (cid:3) Proof of Theorem 3.
Similarly, as in the proof of Theorems 1 and 2 we assume that α > . The case α = 0 was proved already in [4]. Combining zh ′ ν ( z ) h ν ( z ) = 1 − ν z J ′ ν ( z ) J ν ( z ) = 1 − X n ≥ zj ν,n − z with [6, Lemma 2.5] z h ′′ ν ( z ) h ′ ν ( z ) = ν ( ν − J ν ( z ) + (3 − ν ) z J ′ ν ( z ) + zJ ′′ ν ( z )2(2 − ν ) J ν ( z ) + 2 z J ′ ν ( z ) = − X n ≥ zβ ν,n − z , where β ν,n stands for the n th positive zero of the Dini function z (2 − ν ) J ν ( z ) + zJ ′ ν ( z ) , it follows that J ( α, h ν ( z )) = (1 − α ) zh ′ ν ( z ) h ν ( z ) + α (cid:18) zh ′′ ν ( z ) h ′ ν ( z ) (cid:19) = 1 + ( α − X n ≥ zj ν,n − z − α X n ≥ zβ ν,n − z . Applying again the inequality (2.1) we have that1 α Re J ( α, h ν ( z )) ≥ α + (cid:18) − α (cid:19) X n ≥ rj ν,n − r − X n ≥ rβ ν,n − r = 1 α J ( α, h ν ( r )) , where | z | = r. Here we used tacitly that for all n ∈ { , , . . . } we have β ν,n ∈ ( j ν,n − , j ν,n ) , which followsimmediately from Dixon’s theorem [17, p. 480], similarly as in the case the roots α ν,n in the proof ofTheorem 2. Note also that the zeros β ν,n are all real when ν > − , see [17, p. 482], and thus the A. BARICZ, H. ORHAN, R. SZ´ASZ/THE RADIUS OF α -CONVEXITY OF BESSEL FUNCTIONS 7 application of the inequality (2.1) is allowed. Thus, for r ∈ (0 , β ν, ) we get inf z ∈ D (0 ,r ) Re J ( α, h ν ( z )) = J ( α, h ν ( r )) . On the other hand, the function r J ( α, h ν ( r )) is strictly decreasing on (0 , β ν, ) since ∂∂r J ( α, h ν ( r )) = ( α − X n ≥ rj ν,n ( j ν,n − r ) − α X n ≥ rβ ν,n ( β ν,n − r ) < α X n ≥ rj ν,n ( j ν,n − r ) − α X n ≥ rβ ν,n ( β ν,n − r ) < ν > − r ∈ (0 , β ν, ) . Here we used again that the zeros j ν,n and β ν,n interlace and for all n ∈ N ,ν > − r < j ν, β ν, we have that j ν,n ( β ν,n − r ) < β ν,n ( j ν,n − r ) . We also have that lim r ց J ( α, h ν ( r )) = 1 > β and lim r ր β ν, J ( α, g ν ( r )) = −∞ , which means that for z ∈ D (0 , r ) we have Re J ( α, h ν ( z )) > β if and only if r is the unique root of J ( α, h ν ( r )) = β, situated in(0 , β ν, ) . Finally, by using again the interlacing inequalities j ν,n − < β ν,n < j ν,n we obtain the inequality ∂∂α J ( α, h ν ( r )) = X n ≥ rj ν,n − r − X n ≥ rβ ν,n − r < , where ν > − , α ≥ r ∈ (0 , β ν, ) . This implies that the function α J ( α, h ν ( r )) is strictly decreasingon [0 , ∞ ) for all ν > − r ∈ (0 , β ν, ) fixed. Consequently, as a function of α the unique root of theequation J ( α, h ν ( r )) = β is strictly decreasing, where β ∈ [0 , , ν > − r ∈ (0 , β ν, ) are fixed. Thus,when α ∈ (0 ,
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Department of Economics, Babes¸-Bolyai University, 400591 Cluj-Napoca, RomaniaInstitute of Applied Mathematics, ´Obuda University, 1034 Budapest, Hungary
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