Local Convexity Properties of Quasihyperbolic Balls in Punctured Space

Lo al Con v exit y Prop erties of Quasih yp erb oli Balls in Puntured Spae Riku Klén 1 Abstrat This pap er deals with lo al on v exit y prop erties of the quasih yp erb oli metri in the puntured spae. W e onsider on v exit y and starlik eness of quasih yp erb oli balls. 2000 Mathematis Sub jet Classiation: 30C65 Key w ords: quasih yp erb oli ball, lo al on v exit y 1 In tro dution The quasihyp erb oli distan e b et w een t w o p oin ts x and y in a prop er sub domain G of the Eulidean spae R n , n ≥ 2 , is dened b y k G ( x, y ) = inf α ∈ Γ xy Z α | dz | d ( z , ∂ G ) , where d ( z , ∂ G ) is the (Eulidean) distane b et w een the p oin t z ∈ G and the b oundary of G and Γ xy is the olletion of all retiable urv es in G joining x and y . Sine its in tro dution b y F.W. Gehring and B.P . P alk a [ 5℄ in 1976, the quasi- h yp erb oli metri has b een widely applied in geometri funtion theory and math- ematial analysis in general, see e.g. [14, 10℄. Quasih yp erb oli geometry has reen tly b een studied b y P . Hästö [3℄ and H. Lindén [6℄. The purp ose of this pap er is to study the metri spae ( G, k G ) and esp eially lo al on v exit y prop erties of quasihyp erb oli b al ls D G ( x, M ) dened b y D G ( x, M ) = { z ∈ G : k G ( x, z ) < M } . In the dimension n = 2 w e all these balls disks and w e often iden tify R 2 with the omplex plane C . M. V uorinen suggested in [15℄ a general question ab out the on v exit y of balls of small radii in metri spaes. Our w ork is motiv ated b y this question and our main result Theorem 1.1 pro vides an answ er in a partiular ase. F or the denition of starlik e domains see Denition 2.9 . Theorem 1.1. 1) F or x ∈ R n \ { 0 } the quasihyp erb oli b al l D R n \{ 0 } ( x, M ) is stritly  onvex for M ∈ (0 , 1] and it is not  onvex for M > 1 . 2) F or x ∈ R n \ { 0 } the quasihyp erb oli b al l D R n \{ 0 } ( x, M ) is stritly starlike with r esp e t to x for M ∈ (0 , κ ] and it is not starlike with r esp e t to x for M > κ , wher e κ is dene d by (4.1 ) and has a numeri al appr oximation κ ≈ 2 . 83297 . 1 Departmen t of Mathematis, Univ ersit y of T urku, FIN-20014, FINLAND e-mail: riku.klenutu., phone: +358 2 333 6013, fax: +358 2 333 6595 1 Theorem 1.1 in the ase n = 2 is illustrated in Figure 1. O. Martio and J. Väisälä [8℄ ha v e reen tly pro v ed that if G is on v ex then D G ( x, M ) is also on v ex for all x ∈ G and M > 0 . Figure 1: Boundaries of quasih yp erb oli disks D R 2 \{ 0 } ( x, M ) with radii M = 1 , M = 2 and M = κ . 2 Quasih yp erb oli balls with large and small radii In this setion w e onsider the b eha vior of quasih yp erb oli balls with large and small radii. Let us dene φ -uniform domains, whi h w ere in tro dued b y M. V uorinen [13 , 2.49℄, and onsider quasih yp erb oli balls with large radii in φ -uniform domains. W e use notation m ( a, b ) = min { d ( a ) , d ( b ) } , where d ( x ) = d ( x, ∂ G ) . Denition 2.1. Let φ : [0 , ∞ ) → [0 , ∞ ) b e a on tin uous and stritly inreasing homeomorphism. Then a domain G ( R n is φ -uniform if k G ( x, y ) ≤ φ  | x − y | m ( x, y )  for all x, y ∈ G . Lemma 2.2. Fix φ , let G b e φ -uniform, x 0 ∈ G and M > 0 . If x ∈ G with m ( x, x 0 ) > | x − x 0 | /φ − 1 ( M ) then x ∈ D G ( x 0 , M ) . Pr o of. Sine φ is a homeomorphism m ( x, x 0 ) > | x − x 0 | /φ − 1 ( M ) implies φ  | x − x 0 | m ( x, x 0 )  < M and sine G is φ -uniform k G ( x, x 0 ) ≤ φ  | x − x 0 | m ( x, x 0 )  < M . Therefore x ∈ D G ( x 0 , M ) . 2 Denition 2.3. Let δ ∈ (0 , 1 ) and r 0 > 0 b e xed and G ⊂ R n b e a b ounded domain. W e sa y that G satises the ( δ, r 0 ) - ondition if for all z ∈ ∂ G and r ∈ (0 , r 0 ] there exists x ∈ B n ( z , r ) ∩ G su h that d ( x ) > δ r . Theorem 2.4. Assume G is a b ounde d φ -uniform domain and satises the ( δ, r 0 ) -  ondition for a xe d δ ∈ (0 , 1) and r 0 > 0 . L et us assume r 1 ∈ (0 , r 0 ) and x x 0 ∈ G and z ∈ ∂ G . Then d  D G ( x 0 , M ) , z  < r 1 for M > φ  | x 0 − z | + r 2 δ r 2  , (2.5) wher e r 2 = min { r 1 , d ( x 0 ) / 2 } . Pr o of. Sine G satises the ( δ, r 0 ) -ondition and r 2 < r 0 w e an  ho ose x ∈ B n ( z , r 2 ) ∩ G with d ( x ) > δ r 2 . No w m ( x 0 , x ) = min { d ( x 0 ) , d ( x ) } = d ( x ) > δ r 2 and | z − x | < r 2 . The inequalit y (2.5) is equiv alen t to δ r 2 > | x 0 − z | + r 2 φ − 1 ( M ) . Sine | z − x | < r 2 and b y the triangle inequalit y | x 0 − z | + r 2 φ − 1 ( M ) > | x 0 − z | + | z − x | φ − 1 ( M ) ≥ | x 0 − x | φ − 1 ( M ) . No w w e ha v e m ( x 0 , x ) > δ r 2 > | x 0 − z | + r 2 φ − 1 ( M ) > | x 0 − x | φ − 1 ( M ) and b y Lemma 2.2 w e ha v e x ∈ G ∩ D G ( x 0 , M ) . Therefore d  D G ( x 0 , M ) , z  ≤ | z − x | < r 2 ≤ r 1 and the laim is lear. Corollary 2.6. L et G ⊂ R n b e a b ounde d φ -uniform domain and let G satisfy the ( δ, r 0 ) - ondition. F or a xe d s ∈ (0 , r 0 ) and x ∈ G ther e exists a numb er M ( s ) suh that G ⊂ D G  x, M ( s )  + B n ( s ) =  y + z : y ∈ D G  x, M ( s )  , | z | < s  . Pr o of. W e  ho ose M ( s ) > max z ∈ ∂ G φ  | x − z | + r δ r  , where r = min { s, d ( x ) / 2 } . By Theorem 2.4 the assertion follo ws. Let us then p oin t out that quasih yp erb oli balls of small radii b eome more and more lik e Eulidean balls when the radii tend to zero. W e shall study the lo al struture of the b oundary of a quasih yp erb oli ball and sho w that the b oundary is round from the inside and annot ha v e e.g. out w ards direted onial parts. 3 Denition 2.7. Let γ b e a urv e in domain G ( R n . If k G ( x, y ) + k G ( y , z ) = k G ( x, z ) for all x, z ∈ γ and y ∈ γ ′ , where γ ′ is the sub urv e of γ joining x and z , then γ is a ge o desi se gment or briey a ge o desi . W e denote a geo desi b et w een x and y b y J k [ x, y ] . Theorem 2.8. F or a pr op er sub domain G of R n , M > 0 and y ∈ ∂ D G ( x, M ) , let J k [ x, y ] b e a ge o desi se gment of the quasihyp erb oli metri joining x and y . F or z ∈ J k [ x, y ] we have B n  z , | z − y | 1 + u  ⊂ D G ( x, M ) , wher e u = | z − y | /d ( z ) . Pr o of. By [4, Lemma 1℄ there exists J k [ x, y ] . By the  hoie of z w e ha v e M = k G ( x, y ) = k G ( x, z ) + k G ( z , y ) and b y the triangle inequalit y for w ∈ D G  z , k G ( z , y )  w e ha v e k G ( x, w ) ≤ k G ( x, z ) + k G ( z , w ) < M . No w D G  z , k G ( z , y )  ⊂ D G ( x, M ) . By [12, page 347℄ B n  z ,  1 − e − k G ( z ,y )  d ( z )  ⊂ D G  z , k G ( z , y )  and therefore B n  z ,  1 − e − k G ( z ,y )  d ( z )  ⊂ D G ( x, M ) . By [5, Lemma 2.1℄ k G ( z , y ) ≥ log  1 + | z − y | d ( z )  and therefore  1 − e − k G ( z ,y )  d ( z ) ≥  1 − d ( z ) d ( z ) + | z − y |  d ( z ) = | z − y | 1 + u for u = | z − y | d ( z ) . No w B n  z , | z − y | 1 + u  ⊂ B n  z ,  1 − e − k G ( z ,y )  d ( z )  and the laim is lear. 4 No w w e ha v e found a Eulidean ball B n ( z , r ) inside the quasih yp erb oli ball D G ( x, M ) with the follo wing prop ert y: r d  z , ∂ D G ( x, M )  → 1 , when z → ∂ D G ( x, M ) . Geometrially this on v ergene means that the b oundary of the quasih yp erb oli ball m ust b e round from the in terior. The b oundary annot ha v e an y one shap ed orners p oin ting out w ards from the ball. Ho w ev er, there an b e orners in the b oundary p oin ting in w ards to the ball. An example in R 2 \ { 0 } is the quasih y- p erb oli disk with M > π . This example is onsidered in more detail in Remark 4.8. Denition 2.9. Let G ⊂ R n b e a domain and x ∈ G . W e sa y that G is starlike with r esp e t to x if ea h line segmen t from x to y ∈ G is on tained in G . The domain G is stritly starlike with r esp e t to x for x ∈ G if G is b ounded and ea h ra y from x meets ∂ G at exatly one p oin t. The follo wing result onsiders starlik eness of quasih yp erb oli balls in starlik e domains. The same result w as indep enden tly obtained b y J. Väisälä [11℄. Theorem 2.10. If G ( R n is a starlike domain with r esp e t to x , then the quasi- hyp erb oli b al l D G ( x, M ) is starlike with r esp e t to x . Pr o of. W e need to sho w that the funtion f ( y ) = k G ( x, y ) is inreasing along ea h ra y from x to ∂ G . T o simplify notation w e ma y assume x = 0 . Let y ∈ G \ { x } b e arbitrary and denote a geo desi segmen t from x to y b y γ . Let us  ho ose an y y ′ ∈ ( x, y ) and denote γ ′ = | y ′ | | y | γ = cγ . Sine G is starlik e with resp et to x the path γ ′ from x to y ′ is in G . Therefore k G ( x, y ′ ) ≤ Z γ ′ | dz | d ( z ) = Z γ c | dz | d ( cz ) . Sine G is starlik e with resp et to x w e ha v e d ( cz ) ≥ cd ( z ) whi h is equiv alen t to c d ( cz ) ≤ 1 d ( z ) . No w k G ( x, y ′ ) ≤ Z γ c | dz | d ( cz ) ≤ Z γ | dz | d ( z ) = k G ( x, y ) and f is inreasing along ea h ra y from x to ∂ G . F or a domain G ⊂ R n and quasih yp erb oli ball D G ( x, M ) , x ∈ G and M > 0 , w e dene the p oints that  an ae t the shap e of D G ( x, M ) to b e the set { z ∈ ∂ G : | z − y | = d ( y ) for some y ∈ D G ( x, M ) } . Let G b e a domain and x x ∈ G and M > 0 . No w b y [12 , page 347℄ w e kno w that D G ( x, M ) ⊂ B n ( x, Rd ( x )) , for R = e M − 1 , and therefore for ea h y ∈ D G ( x, M ) w e ha v e d ( y ) ≤ d ( x ) + 2 Rd ( x ) = d ( x )(2 e M − 1) . This fat is generalized in the follo wing lemma. 5 Lemma 2.11. L et G ( R n b e a domain, x ∈ G and y ∈ ∂ G . Then the p oints that  an ae t the shap e of the quasihyp erb oli b al l D G ( x, M ) for M ∈ (0 , 1] ar e in the losur e of the set U y = B n  x, | x − y | (2 e M − 1)  \ { z ∈ R n \ { y } : ∡ x ′ y z ≤ π / 2 − 1 , x ′ = 2 y − x } , wher e ∡ x ′ y z is the angle b etwe en line se gments [ x ′ , y ] and [ z , y ] at y . Pr o of. Let us onsider G ′ = R n \ { y } . No w G ⊂ G ′ and therefore D G ( x, M ) ⊂ D G ′ ( x, M ) . No w the p oin ts that an aet the shap e of D G ( x, M ) need to b e inside B n  x, | x − y | (2 e M − 1)  . Let z ∈ ∂ D G ′ ( x, M ) . Beause M ≤ 1 w e ha v e b y (3.1) ∡ xy z ≤ 1 . Therefore the p oin ts in { z ∈ R n \ { y } : ∡ x ′ y z ≤ π / 2 − 1 , x ′ = 2 y − x } do not aet the shap e of D G ′ ( x, M ) . Sine D G ( x, M ) ⊂ D G ′ ( x, M ) , the laim is lear. Theorem 2.12. F or a domain G ( R n , M ∈ (0 , 1] and x ∈ G the quasihyp erb oli b al l D G ( x, M ) is starlike with r esp e t to x . Pr o of. W e denote V x = G ∩ \ y ∈ ∂ G U y ! . The set V x on tains all of the b oundary p oin ts of G that aet the shap e of D G ( x, M ) . Therefore for xed x ∈ G w e ha v e D G ( x, M ) = D V x ( x, M ) and D G ( x, M ) is starlik e with resp et to x b y Theorem 2.10 , b eause V x is starlik e with resp et to x . Remark 2.13. In Lemma 2.11 and Theorem 2.12 w e ould replae M ∈ (0 , 1] b y M ∈ (0 , α ] and ∡ x ′ y z ≤ π / 2 − 1 b y ∡ x ′ y z ≤ π / 2 − α for an y α ∈ [1 , π / 2) . This mo died v ersion of Theorem 2.12 w as also pro v ed b y J. Väisälä [10 , Theorem 3.11℄. 3 Con v exit y of quasih yp erb oli balls in puntured spae The set R n \ { z } , z ∈ R n , is alled a puntured spae. T o simplify notation w e ma y assume z = 0 . In this setion w e will nd v alues M su h that the quasih yp erb oli ball D R n \{ 0 } ( x, M ) is on v ex for all x ∈ R n \ { 0 } . Let us assume that x, y ∈ R n \ { 0 } and that the angle ϕ b et w een segmen ts [0 , x ] and [0 , y ] satises 0 < ϕ ≤ π . It an b e sho wn [7, page 38℄ that k R n \{ 0 } ( x, y ) = s ϕ 2 + log 2 | x | | y | . (3.1) In partiular, w e see that k R n \{ 0 } ( x, y ) = k R n \{ 0 } ( x, y 1 ) , where y 1 is obtained from y b y the in v ersion with resp et to S n − 1 ( | x | ) , i.e. y 1 = y | x | 2 / | y | 2 . Hene this 6 in v ersion maps the quasih yp erb oli sphere { z ∈ R n \ { 0 } : k R n \{ 0 } ( x, z ) = M } on to itself. Quasih yp erb oli balls are similar in R n \ { 0 } for xed M . In other w ords an y quasih yp erb oli ball of radius M an b e mapp ed on to an y other quasih yp erb oli ball of radius M b y rotation and stret hing. W e will rst onsider on v exit y of the quasih yp erb oli disks in the puntured plane R 2 \ { 0 } and then extend the results to the puntured spae R n \ { 0 } . By (3.1 ) w e ha v e a o ordinate represen tation in the ase n = 2 x = ( | x | cos ϕ, | x | sin ϕ ) =  e ± √ M 2 − ϕ 2 cos ϕ, e ± √ M 2 − ϕ 2 sin ϕ  , (3.2) for x ∈ ∂ D R 2 \{ 0 } (1 , M ) and − M ≤ ϕ ≤ M . By using this presen tation w e will pro v e the follo wing result. Theorem 3.3. F or M > 1 and z ∈ R 2 \ { 0 } the quasihyp erb oli disk D R 2 \{ 0 } ( z , M ) is not  onvex. Pr o of. W e ma y assume z = 1 and let x ∈ ∂ D R 2 \{ 0 } ( z , M ) b e arbitrary . Assume M > 1 . By ( 3.2 ) w e ha v e x =  e ± √ M 2 − ϕ 2 cos ϕ, e ± √ M 2 − ϕ 2 sin ϕ  , where − M ≤ ϕ ≤ M . If M > π / 2 , then the laim is lear b y symmetry b eause Re x = e − M > 0 for ϕ = 0 and Re x < 0 for ϕ = ± M . W e will sho w that the funtion f ( ϕ ) = e − √ M 2 − ϕ 2 cos ϕ is ona v e in the neigh b orho o d of ϕ = 0 and the funtion g ( ϕ ) = e − √ M 2 − ϕ 2 sin ϕ is inreasing in  0 , min { M , π 2 }  . This will imply non-on v exit y of D R 2 \{ 0 } ( z , M ) . First, g ′ ( ϕ ) = e − √ M 2 − ϕ 2 cos ϕ + ϕ sin ϕ p M 2 − ϕ 2 ! and this is learly non-negativ e for 0 < ϕ < min { M , π 2 } . Therefore g ( ϕ ) is inreas- ing. Seond, b y a straigh tforw ard omputation w e obtain f ′ ( ϕ ) = e − √ M 2 − ϕ 2 ϕ cos ϕ p M 2 − ϕ 2 − sin ϕ ! and f ′′ ( ϕ ) = e − √ M 2 − ϕ 2  M 2 − p M 2 − ϕ 2 ( M 2 − 2 ϕ 2 )  cos ϕ + 2 ϕ ( ϕ 2 − M 2 ) sin ϕ ( p M 2 − ϕ 2 ) 3 . No w f ′ (0) = 0 and f ′′ (0) = e − M (1 / M − 1 ) < 0 and therefore f ( ϕ ) is ona v e in the neigh b orho o d of ϕ = 0 . 7 Theorem 3.3 an easily b e extended to the ase n ≥ 3 . Corollary 3.4. If M > 1 and z ∈ R n \ { 0 } , then the quasihyp erb oli b al l D R n \{ 0 } ( z , M ) is not  onvex. Pr o of. Let us  ho ose an y y ∈ R n \ { 0 } su h that y 6 = t z for all t ∈ R . No w D R n \{ 0 } ( z , M ) ∩ span (0 , y , z ) is not on v ex b y Theorem 3.3 and therefore the quasi- h yp erb oli ball D R n \{ 0 } ( z , M ) annot b e on v ex. Let us no w onsider the on v exit y of the quasih yp erb oli balls in the ase M ≤ 1 and n = 2 . Theorem 3.5. F or 0 < M ≤ 1 and z ∈ R 2 \ { 0 } the quasihyp erb oli disk D R 2 \{ 0 } ( z , M ) is stritly  onvex. Pr o of. Let z = 1 and x ∈ ∂ D R 2 \{ 0 } ( z , M ) . By symmetry it is suien t to onsider the upp er half D of ∂ D R 2 \{ 0 } ( z , M ) , whi h is giv en b y x = x ( s ) = ( e s cos ϕ, e s sin ϕ ) , (3.6) where M ∈ (0 , π ) , s ∈ [ − M , M ] and ϕ = ϕ ( s ) = √ M 2 − s 2 . No w ϕ ′ ( s ) = − s/ϕ ( s ) and therefore for s ∈ ( − M , M ) x ′ ( s ) = e s ϕ ( s )  a ( s ) , b ( s )  , where a ( s ) = ϕ ( s ) cos ϕ ( s ) + s sin ϕ ( s ) and b ( s ) = ϕ ( s ) sin ϕ ( s ) − s cos ϕ ( s ) . No w t ( s ) =  a ( s ) , b ( s )  is a tangen t v etor of D for s ∈ [ − M , M ] . Equalit y t ( s ) = 0 is equiv alen t to s 2 = − ϕ ( s ) 2 , whi h nev er holds. Sine t ( s ) 6 = 0 for all s ∈ [ − M , M ] the angle α ( s ) = arg t ( s ) is a on tin uous funtion on ( − M , M ) . W e need to sho w that α ( s ) is stritly dereasing on [ − M , M ] . Sine α ( s ) = arctan  b ( s ) /a ( s )  and arctan is stritly inreasing, w e need to sho w that c ( s ) = b ( s ) /a ( s ) is stritly dereasing. By a straigh tforw ard omputation c ′ ( s ) = a ( s ) b ′ ( s ) − b ( s ) a ′ ( s ) a ( s ) 2 = − (1 + s ) M 2 ϕ ( s ) a ( s ) 2 (3.7) and the assertion follo ws. Remark 3.8. The b oundary ∂ D R 2 \{ 0 } (1 , M ) is smo oth sine α ( s ) is on tin uous, t ( M ) = (0 , − M ) and t ( − M ) = (0 , M ) . By using the symmetry of the quasih yp erb oli balls w e an extend Theorem 3.5 to the ase of puntured spae. Lemma 3.9. L et the domain G ⊂ R n b e symmetri ab out a line l , G ∩ l 6 = ∅ and G ∩ L b e stritly  onvex for any plane L with l ⊂ L . Then G is stritly  onvex. 8 Pr o of. W e ma y assume that the line l is the rst o ordinate axis of R n to simplify notation. Let us dene funtion f : R → [0 , ∞ ) b y f ( x ) =  d ( x, z ) , if there exists z = ( x, z 2 , . . . , z n ) ∈ ∂ G 0 , otherwise. Sine G is symmetri ab out l and G ∩ l 6 = ∅ there exists su h x 0 , x 1 ∈ R that f [ x 0 , x 1 ] = [0 , d ] for d < ∞ and f ( x 0 ) = 0 = f ( x 1 ) . Sine G ∩ L is on v ex the funtion f is ona v e on [ x 0 , x 1 ] . Let x, y ∈ G , x 6 = y b e arbitrary and denote A x = { z = ( x 1 , z 2 , . . . , z n ) ∈ G : d ( z , l ) = d ( x, l ) } and A y = { z = ( y 1 , z 2 , . . . , z n ) ∈ G : d ( z , l ) = d ( y , l ) } . The line segmen t [ x, y ] is on tained in the losure of the on v ex h ull of A x ∪ A y , whi h is on tained in G b y the ona vit y of f . Corollary 3.10. F or 0 < M ≤ 1 and z ∈ R n \ { 0 } the quasihyp erb oli b al l D R n \{ 0 } ( z , M ) is stritly  onvex. Pr o of. By (3.1) the quasih yp erb oli ball D R n \{ 0 } ( x, M ) is symmetri ab out the line that on tains x and 0 . By Lemma 3.9 and Theorem 3.5 D R n \{ 0 } ( x, M ) is stritly on v ex for 0 < M ≤ 1 . 4 Starlik eness of quasih yp erb oli balls in puntured spae In this setion w e will nd the maxim um v alue of the radius M for whi h the quasih yp erb oli ball D R n \{ 0 } ( x, M ) is starlik e with resp et to x . As in the previous setion w e will rst onsider the quasih yp erb oli disks in the puntured plane and then extend the results to the puntured spae. Let us dene a onstan t κ as the solution of the equation cos p p 2 − 1 + p p 2 − 1 sin p p 2 − 1 = e − 1 (4.1) for p ∈ [1 , π ] . The pro of of the next theorem sho ws that the equation (4.1 ) has only one solution κ on [1 , π ] with n umerial appro ximation κ ≈ 2 . 83297 . Remark 4.2. A ording to [1℄ the n um b er κ w as rst in tro dued b y P .T. Mo an u in 1960 [9℄. Later V. Anisiu and P .T. Mo an u sho w ed [ 1, page 99℄ that if f is an analyti funtion in the unit disk, f (0) = 0 and     f ′′ ( z ) f ′ ( z )     ≤ κ, then f is starlik e with resp et to 0. Theorem 4.3. The quasihyp erb oli disk D R 2 \{ 0 } ( x, M ) is stritly starlike with r e- sp e t to x for 0 < M ≤ κ and is not starlike with r esp e t to x for M > κ . 9 Pr o of. Beause of symmetry w e will onsider ∂ D R 2 \{ 0 } ( x, M ) only ab o v e the real axis and b y the similarit y it is suien t to onsider only the ase x = 1 . By Theorem 3.5 w e need to onsider M ∈ (1 , π ) . Let us denote b y l ( s ) a tangen t line of the upp er half of ∂ D R 2 \{ 0 } (1 , M ) . The slop e of the tangen t line l ( s ) is desrib ed b y the funtion c ( s ) dened in the pro of of Theorem 3.5 . By (3.7 ) the funtion c ( s ) is inreasing on [ − M , − 1] and dereasing on [ − 1 , M ] . W e need to nd M su h that l ( s ) , s ∈ [ − M , M ] , go es through p oin t 1 exatly one. In other w ords, w e need to nd M su h that l ( − 1) go es through 1. The tangen t line l ( s ) go es through 1 if and only if c ( s ) = x 2 x 1 − 1 , (4.4) where x 1 = e s cos ϕ ( s ) and x 2 = e s sin ϕ ( s ) . The equation (4.4) in the sp eial ase s = − 1 is equiv alen t to e cos √ M 2 − 1 + e √ M 2 − 1 sin √ M 2 − 1 − 1 ( e − cos √ M 2 − 1)( √ M 2 − 1 cos √ M 2 − 1 − sin √ M 2 − 1) = 0 , whi h holds if and only if M = κ . W e will nally sho w that M = κ is the only solution of ( 4.1) on (1 , π ) . W e de- ne funtion h ( x ) = cos x + x sin x − e − 1 and sho w that it has only one ro ot on (0 , √ π 2 − 1) . Sine h ′ ( x ) = x cos x , h (0)1 − e − 1 > 0 and h ( √ π 2 − 1) < h (11 π / 12) < 0 the funtion h has only one ro ot on (0 , √ π 2 − 1) and the asser- tion follo ws. Corollary 4.5. The quasihyp erb oli b al l D R n \{ 0 } ( x, M ) is stritly starlike with r esp e t to x for 0 < M ≤ κ and is not starlike with r esp e t to x for M > κ . Pr o of. By Theorem 4.3 the laim is true for n = 2 . Let us assume n > 2 and  ho ose x ∈ R n \ { 0 } and M ∈ (0 , κ ] . Let us assume, on the on trary , that there exist y ∈ ∂ D R n \{ 0 } ( x, M ) and z ∈ ( x, y ) su h that z ∈ ∂ D R n \{ 0 } ( x, M ) . No w z ∈ ∂ D R n \{ 0 } ( x, M ) ∩ span (0 , x, y ) and therefore D R 2 \{ 0 } ( x, M ) is not stritly starlik e with resp et to x . This is a on tradition b y Theorem 4.3 . Remark 4.6. Let us onsider the starlik eness prop ert y of the quasih yp erb oli disk D R 2 \{ 0 } ( x, M ) with resp et to an y p oin t z ∈ D R 2 \{ 0 } ( x, M ) . F or M > 1 and z = ( e − M + ε ) x/ | x | , where ε > 0 , w e an  ho ose ε so small that D R 2 \{ 0 } ( x, M ) is not starlik e with resp et to z . On the other hand for M < λ ≈ 2 . 96489 84 , where λ is a solution of cos p p 2 − 1 + p p 2 − 1 sin p p 2 − 1 = e − 1 − p , (4.7) D R 2 \{ 0 } ( x, M ) is starlik e with resp et to z = ( e M − ε ) x/ | x | for small enough ε > 0 . This is also true for quasih yp erb oli balls D R n \{ 0 } ( x, M ) . The equation ( 4.7 ) an b e obtained b y similar omputations as in the pro of of Theorem 4.3. Remark 4.8. F or M ≤ π w e note that lim ϕ → M c ( s ) = −∞ and lim ϕ →− M c ( s ) = ∞ 10 and therefore D R n \{ 0 } ( x, M ) smo oth. F or M > π the b oundary ∂ D R n \{ 0 } ( x, M ) is dened b y (3.6 ) for s ∈ [ m, M ] , where m = max { t ∈ ( − M , M ) : sin √ M 2 − t 2 = 0 } . Therefore lim ϕ → M c ( s ) = −∞ and lim ϕ → m c ( s ) = − m cos ϕ ( m ) ϕ ( m ) cos ϕ ( m ) = − m ϕ ( m ) , where | − m/ϕ ( m ) | < ∞ , and D R n \{ 0 } ( x, M ) is not smo oth at  e m sin ϕ ( m ) , 0  . Note that b y (3.1) D R 2 \{ 0 } ( x, M ) is not simply onneted for M > π and is simply onneted for M ∈ (0 , π ] . Pr o of of The or em 1.1. The laim is lear b y Corollaries 3.4 , 3.10 and 4.5. The follo wing lemma sho ws a prop ert y of the Eulidean radius of a quasih y- p erb oli ball. Lemma 4.9. L et M ∈ (0 , κ ] , z ∈ R n \ { 0 } and x, y ∈ ∂ D R n \{ 0 } ( z , M ) . Then ∡ xz 0 < ∡ y z 0 implies | x − z | < | y − z | . Pr o of. Sine M ≤ κ the quasih yp erb oli ball D R n \{ 0 } ( z , M ) is stritly starlik e with resp et to z b y Theorem 4.5 and the angle ∡ xz 0 determines the p oin t x uniquely . By symmetry and similarit y it is suien t to onsider only the ase n = 2 and z = 1 . W e will sho w that the funtion f ( s ) = | x ( s ) − 1 | 2 is stritly inreasing on ( − M , M ) , where x ( s ) dened b y (3.6 ). No w f ( s ) = | x ( s ) | 2 + 1 − 2 | x ( s ) | cos p ( s ) = e 2 s + 1 − 2 e s cos ϕ ( s ) for s ∈ [ − M , M ] and f ′ ( s ) = 2 e s  e s − cos ϕ ( s ) − s sin ϕ ( s ) ϕ ( s )  . If s ∈ (0 , M ) , then e s − cos ϕ ( s ) − s sin ϕ ( s ) ϕ ( s ) ≥ e s − cos ϕ ( s ) − s ≥ e s − 1 − s > 0 and f ′ ( s ) > 0 . If s ∈ [ − M , 0) , then e s − cos ϕ ( s ) − s sin ϕ ( s ) /ϕ ( s ) > 0 is equiv alen t to e − t − cos ϕ ( t ) + t sin ϕ ( t ) /ϕ ( t ) > 0 for t ∈ (0 , M ] . Beause M < 3 , b y elemen tary alulus e − t − cos ϕ ( t ) + t sin ϕ ( t ) ϕ ( t ) ≥  1 − t + t 2 2 − t 3 6  −  1 − ϕ ( t ) 2 2 + ϕ ( t ) 4 24  +  t − t ϕ ( t ) 2 6  = 1 24  12 M 2 − M 4 − 4 M 2 t + 2 M 2 t 2 − t 4  > 0 and also f ′ ( s ) > 0 . Therefore f is stritly inreasing and the assertion follo ws. Finally w e p ose an op en problem onerning the uniqueness of short geo desis: are quasih yp erb oli geo desis with length less than π alw a ys unique? A know le dgements. This pap er is part of the author's PhD thesis, urren tly written under the sup ervision of Prof. M. V uorinen and supp orted b y the A adem y of Finland pro jet 8107317. 11 Referenes [1℄ V. Anisiu, P.T. Moanu : On a simple suient  ondition for starlike- ness. Mathematia (Cluj) 31 (54) (1989), 97101. [2℄ R.H. F o wler : The Elementary Dier ential Ge ometry of Plane Curves. Cam bridge Univ ersit y Press, 1929. [4℄ F.W. Gehring, B.G. 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