Analytic representation of the generalized Pascal snail and its applications
AANALYTIC REPRESENTATION OF THE GENERALIZED PASCALSNAIL AND ITS APPLICATIONS
S. KANAS AND V. S. MASIH Abstract.
We find an unifying approach to the analytic representation of the domainbounded by a generalized Pascal snail. Special cases as Pascal snail, Both leminiscate,conchoid of the Sluze and a disc are included. The behavior of functions related to generalizedPascal snail are demonstrated. The analytic representation of a Pascal snail
For − ≤ α ≤ − ≤ β ≤ αβ (cid:54) = ±
1, and 0 ≤ γ < L α,β,γ denote the complexvalued mapping(1.1) L α,β,γ ( z ) = (2 − γ ) z (1 − αz ) (1 − βz ) = ∞ (cid:88) n =1 B n z n = (2 − γ ) ∞ (cid:88) n =1 (cid:18) α n − β n α − β (cid:19) z n , α (cid:54) = β ;(2 − γ ) ∞ (cid:88) n =1 nα n − z n , α = β, where z ∈ D = { z ∈ C : | z | < } . We note that L α,β,γ maps D onto a domain D ( α, β, γ )whose boundary is a given by ∂ D ( α, β, γ ) = (cid:40) w = u + iv : (cid:0) − γ ) u + ( α + β )( u + v ) (cid:1) (1 + αβ ) + 4(1 − γ ) v (1 − αβ ) − ( u + v ) = 0 (cid:41) . Indeed, for z = e iθ , with θ ∈ [0 , π ), we obtain L ( z ) := z (1 − αz ) (1 − βz ) = e iθ (cid:0) − αe − iθ (cid:1) (cid:0) − βe − iθ (cid:1) | − αe iθ | | − βe iθ | = (1 + αβ ) cos θ − ( α + β ) + i (1 − αβ ) sin θ (1 + α − θ )(1 + β − β cos θ ) , − ≤ α, β ≤ . (1.2)Let u = u ( θ ) = (cid:60) (cid:8) L (cid:0) e iθ (cid:1)(cid:9) and v = v ( θ ) = (cid:61) (cid:8) L (cid:0) e iθ (cid:1)(cid:9) . Then(1.3) u = (1 + αβ ) cos θ − ( α + β )(1 + α − α cos θ )(1 + β − β cos θ ) , v = (1 − αβ ) sin θ (1 + α − α cos θ )(1 + β − β cos θ ) . Hence, u, v satisfy the equation(1.4) (cid:0) u + ( α + β )( u + v ) (cid:1) (1 + αβ ) + v (1 − αβ ) − ( u + v ) = 0 ( αβ (cid:54) = ± . Mathematics Subject Classification.
Key words and phrases. domain bounded by the generalized Pascal snail, Booth leminiscate, conchoid ofthe Sluze, Pascal snail, univalent functions, applied mathematics, starlike and convex functions. Corresponding author. a r X i v : . [ m a t h . C V ] M a r S. KANAS AND V. S. MASIH
Therefore L α,β,γ maps the unit circle onto a curve (cf. [4, 5])(1.5) (cid:0) − γ ) u + ( α + β )( u + v ) (cid:1) (1 + αβ ) + 4(1 − γ ) (1 − αβ ) v = ( u + v ) ( αβ (cid:54) = ± , or(1.6) (cid:18) u + v − − γ )( α + β )(1 − α )(1 − β ) u (cid:19) = 4(1 − γ ) (1 + αβ ) (1 − α ) (1 − β ) u + 4(1 − γ ) (1 + αβ ) (1 − α )(1 − β )(1 − αβ ) v , that is generalization of the Pascal snail (see Fig. 1 and Fig 2).The wide applications of the Pascal snail have been known since their description; thenewest ones rely on the application to figure the path of airflow around object like planewings, in the design of race and train tracks but also in cryptography for selecting the pointsof the curve (ellipse, leminiscate, etc.) over the prime fields. Also, the leminiscate are used inthe construction of grids on irregular regions in the development of software for numericallysolving partial differential equations. Very recently a method based on leminiscate is appliedfor meander like regions and rely on covering the region with sectors bounded by two confocalleminiscate and two arcs orthogonal to the Pascal snail (cf. [7]).In this paper we will deal with the Pascal snail (1.5) or (1.6) and its analytical represen-tation. Also, we will discuss the special cases of (1.5) or (1.6) which give some interestingcurves. Let us consider individual cases separately. By a symmetry, from now on we make - - uv (a) α = − . , β = 0 . , γ = 0 . - - - - - - - uv (b) α = − . , β = 0 . , γ = 0 . Figure 1.
The image of D under L α,β,γ ( z )the assumption: β ≥ α , unless otherwise stated.1.1. Circular domains.
We get a circle for the case, when one of the parameter α or β iszero, and the second is in the interval ( − , α = 0 < β <
1. Then L ,β,γ has the form L ,β,γ ( z ) = 2(1 − γ ) z − β z , and L ,β,γ ( D ) is a circular domain D (0 , β, γ ) = (cid:26) w ∈ C : (cid:12)(cid:12)(cid:12)(cid:12) w − γ ) − β − β (cid:12)(cid:12)(cid:12)(cid:12) < − β (cid:27) . For the case α = β = 0 a curve D (0 , , γ ) is a circle | w | < − γ ). NALYTIC REPRESENTATION OF THE GENERALIZED PASCAL SNAIL AND ITS APPLICATIONS 3 - - - - - - - uv (a) α = − . , β = − . , γ = 0 . - - uv (b) α = 0 . , β = 0 . , γ = 0 . Figure 2.
The image of D under L α,β,γ ( z )1.2. Halfplane.
For the case when α = 0 and β = 1 the domain L ( D ) is the halfplane (cid:60) w > γ −
1. The case β = 0 , α = − (cid:60) w < − γ which is not the ones ofinterest to us.1.3. Pascal snail regions.
In the case β = α ∈ ( − , \ { } the function L α,α,γ becomes(1.7) L α,α,γ ( z ) = 2(1 − γ ) z (1 − αz ) , with 0 ≤ γ <
1, that maps the unit disk onto simply connected and bounded region, whichcan be described as(1.8) ( u + v − eau ) = a ( u + v ) , where e = 2 α α , a = 2(1 − γ )(1 + α )(1 − α ) . The equation (1.8) can be rewritten in a polar equation(1.9) L α,α,γ ( e it ) = (cid:8) ρ e i ϕ : ρ = Θ( α, ϕ ) , − π < ϕ ≤ π (cid:9) , where Θ( α, ϕ ) = 2(1 − γ ) | α | α α + 2 | α | cos ϕ (1 − α ) . The boundary curve, known as Pascal snail (lima¸con of Pascal), is a bicircular rationalplane algebraic curve of degree 4 which belongs to the family of curves called centered tro-choids or epitrochoids (cf. Fig. 3. Certainly L , ,γ ( D ) is a disk). Pascal snail is the inversionof conic sections with respect to a focus.We note that | e | < α (cid:54) = 1. In this case the snail is elliptic which is inverse of anellipse with respect to its focus. In the case, when | e | < /
2, that is α ∈ (cid:0) − √ , − √ (cid:1) ,the domain bounded by the Pascal snail (1.8) is convex, and tends to the circle when α →
0. For | e | = 1 / | e | > / α ∈ (cid:0) − , − √ (cid:1) ∪ (cid:0) − √ , (cid:1) , the curve has a shape of a bean. The casewhen the Pascal snail has a loop does not hold, because it is equivalent to the inequality(1 − α ) <
0. Summarizing, the domain bounded by the Pascal snail is bounded, convex for
S. KANAS AND V. S. MASIH - - - uv (a) α = 2 − √ , γ = 0 . - - - - - - uv (b) α = − √ , γ = 0 . Figure 3.
The image of D under L α,α,γ ( z ) α ∈ (cid:2) − √ , − √ (cid:3) , concave for α ∈ (cid:0) − , − √ (cid:1) ∪ (cid:0) − √ , (cid:1) , and symmetric withrespect to real axis.The function L α,α,γ ( z ) with α (cid:54) = 0 can also be written as a composition of two analyticunivalent functions, that is, L α,α,γ ( z ) = ( h ◦ h ) ( z ) = 1 − γ α (cid:34)(cid:18) αz − αz (cid:19) − (cid:35) , where h ( z ) = 1 + αz − αz and h ( z ) = 1 − γ α (cid:0) z − (cid:1) . The function h is univalent in D and h is univalent in h ( D ) = (cid:110) w ∈ C : (cid:12)(cid:12)(cid:12) w − w +1 (cid:12)(cid:12)(cid:12) < | α | (cid:111) .1.4. Conchoid of the Sluze.
In the third special case we set β = 1 , α ∈ ( − , \ { } or α = − , β ∈ ( − , \ { } . Let us consider β = 1 , α ∈ ( − , L α, ,γ has a form(1.10) L α, ,γ ( z ) = 2(1 − γ ) z (1 − αz )(1 − z ) , where 0 ≤ γ <
1, that maps the unit disk onto simply connected region with boundary thatis a curve ∂ D ( α, , γ ) = (cid:26) u + i v : (cid:18) u + (1 + α )(1 − γ )(1 − α ) (cid:19) ( u + v ) − α (1 − γ )(1 − α ) (1 + α ) u = 0 (cid:27) = (cid:40) u + iv : (cid:2) − γ ) u + (1 + α )( u + v ) (cid:3) (1 + α ) + 4(1 − γ ) v (1 − α ) = (cid:2) u + v (cid:3) (cid:41) known as the Conchoid of de Sluze, see Fig. 4. We note that the special case β = 1 and − < α < α = − , β ∈ ( − , \ { } we obtain the conchoid of de Sluze (see Fig. 5)of the form NALYTIC REPRESENTATION OF THE GENERALIZED PASCAL SNAIL AND ITS APPLICATIONS 5 - - - - - - uv (a) α = − . , γ = 0 . - - - - - - uv (b) α = 0 . , γ = 2 / Figure 4.
The image of D under L α, ,γ - - - - - uv (a) β = − . , γ = 2 / - - - - - - uv (b) β = γ = 0 . Figure 5.
The image of D under L − ,β,γ ∂ D ( − , β, γ ) = (cid:26) u + i v : (cid:18) u − (1 − γ )(1 − β )(1 + β ) (cid:19) ( u + v ) − β (1 − γ )(1 + β ) (1 − β ) u = 0 (cid:27) = (cid:40) u + iv : (cid:2) − γ ) u − (1 − β )( u + v ) (cid:3) (1 − β ) + 4(1 − γ ) v (1 + β ) = (cid:2) u + v (cid:3) (cid:41) , symmetric to the ∂ D ( α, , γ ) with respect to the imaginary axis.1.5. Hippopede. Leminiscate of Booth.
Here we let β = − α, α ∈ ( − , L α, − α,γ is of the form(1.11) L α, − α,γ ( z ) = 2(1 − γ ) z − α z , S. KANAS AND V. S. MASIH and the equation (1.5) or (1.6) reduces to ( u + v ) = c u + d v , with c = 2(1 − γ ) / (1 − α ) , d = 2(1 − γ ) / (1 + α ), that is(1.12) ∂ D ( α, − α, γ ) = (cid:40) u + iv : 4(1 − γ ) (1 − α ) u + 4(1 − γ ) (1 + α ) v = (cid:0) u + v (cid:1) (cid:41) . We remind that the hippopede is the bicircular rational algebraic curve of degree 4, symmetricwith respect to both axes. Any hippopede is the intersection of a torus with one of its tangentplanes that is parallel to its axis of rotational symmetry. When c > d > α (cid:54) = 0)such a curve is known as an oval or leminiscate of Booth, see Fig. 6. Since the case d = − c does not hold, the leminiscate (1.12) do not reduce to the leminiscate of Bernoulli. - - - - - - uv (a) α = − (cid:112) / , γ = 0 . Figure 6.
The image of D under L α, − α,γ ( z )We note that for c/ √ < d < c √
2, the domain bounded by the hippopede is convex, thatis for − (cid:112) − √ < α <
0, and the curve is called Booth’s oval. For d = c/ √
2, that isfor α = − (cid:112) − √ α ∈ (cid:104) − (cid:112) − √ , (cid:17) , andconcave for α ∈ (cid:16) − , − (cid:112) − √ (cid:17) .1.6. Remaining cases.
In this case, we consider the remaining range of parameters, i.e. − < α < β <
1, which were not considered in previous subsections. In these cases a curve L α,β,γ ( e it ) is the generalized Pascal snail, that has the form(1.13) ( u + v − au ) = c u + d v with(1.14) a = 2(1 − γ )( α + β )(1 − α )(1 − β ) , c = 2(1 − γ )(1 + αβ )(1 − α )(1 − β ) , d = 2(1 − γ )(1 + αβ )(1 − αβ ) (cid:112) (1 − α )(1 − β ) . We note, that the curve represented by the equation (1.13) has similar properties to thePascal snail, and is symmetric only with respect to real axis. It has either horizontal eight-like shape, bean-shape, pear-shape or is convex. From this reason the region bounded by(1.13) is convex, or concave. As we can see in the Theorem 1.1 the minimum and maximumof real part are not always achieved on the real axis. Taking into account the geometricalproperties of set L ( D ), we get the following. NALYTIC REPRESENTATION OF THE GENERALIZED PASCAL SNAIL AND ITS APPLICATIONS 7
Theorem 1.1 ([5]) . Let − ≤ α ≤ β ≤ and αβ (cid:54) = ± . Then max ≤ θ< π (cid:60) L ( e iθ ) = (1+ αβ ) − αβ )[2 √ αβ (1 − α )(1 − β ) − ( α + β )(1 − αβ )] for ( α, β ) ∈ B , − α )(1 − β ) otherwise , min ≤ θ< π (cid:60) L ( e iθ ) = − (1+ αβ ) − αβ )[2 √ αβ (1 − α )(1 − β )+( α + β )(1 − αβ )] for ( α, β ) ∈ B , − α )(1+ β ) otherwise , where B = { < α < , β ( α ) < β < } , B = {− < α < , α < β < β ( α ) } , with β ( α ) = (1 + α ) √ α + 14 α + 1 − ( α + 6 α + 1)2 α (1 − α ) ,β ( α ) = (1 − α ) √ α − α + 1 − α + 6 α − α (1 + α ) . The sets B , B are represented on a Fig. 7. - - + β ( α ) = αβ ( α ) = α β ( α ) β ( α ) β = αβ = - α B B - - - - αβ ( α ) Figure 7.
The range of the parameters α, β . In the special cases we have max ≤ θ< π (cid:60) L ( e iθ ) = − α (cid:16) α − α (cid:17) for − < α ≤ √ − , β = α, − α ) for √ − ≤ α < , β = α, − α − α ) for − < α ≤ , β = 1 , − α ) for ≤ α < , β = 1 , − α for − < α < , β = − α. min ≤ θ< π (cid:60) L ( e iθ ) = − α (cid:16) α − α (cid:17) for − √ ≤ α < , β = α, − α ) for − < α ≤ √ − , β = α, − α ) for − < α ≤ , β = 1 , − − α α ) for ≤ α < , β = 1 , − − α for − < α < , β = − α. S. KANAS AND V. S. MASIH
Conclusions.
In general L ( D ) is a domain symmetric about the real axis and starlikewith respect to origin and such that L (0) = 0, L (cid:48) (0) = 1 >
0. The geometrical propertiesof the regions L ( D ) provides a natural bridge between the convex and concave domains. Wealso note that such domains were discussed in relation of generalized typically-real functionsand generalized Chebyshev polynomials of the second kind [4, 5].From Theorem 1.1 we conclude the following Corollary. Corollary 1.2.
Let − ≤ α ≤ β ≤ , αβ (cid:54) = ± and ≤ γ < , and let L α,β,γ be the functiondefined by (1.1) . Then, for z ∈ D we have (cid:60) { L α,β,γ ( z ) } > L ( α, β, γ ) = − (1+ αβ ) (1 − γ )(1 − αβ )[2 √ αβ (1 − α )(1 − β )+( α + β )(1 − αβ )] for ( α, β ) ∈ B , − − γ )(1+ α )(1+ β ) otherwise, and (cid:60) { L α,β,γ ( z ) } < M ( α, β, γ ) = (1+ αβ ) (1 − γ )(1 − αβ )[2 √ αβ (1 − α )(1 − β ) − ( α + β )(1 − αβ )] for ( α, β ) ∈ B , − γ )(1 − α )(1 − β ) otherwise, Also, if α = 0 , then (cid:12)(cid:12)(cid:12)(cid:12) L ,β,γ ( z ) − − γ ) β − β (cid:12)(cid:12)(cid:12)(cid:12) ≤ − γ )1 − β . In the sequel we will use the following lemma.
Lemma 1.3 ([1]) . Let z is a complex number with positive real part. Then for any realnumber t such that t ∈ [0 , , we have (cid:60) (cid:8) z t (cid:9) ≥ ( (cid:60) z ) t . Subclass of the Carath`eodory class related to the generalized Pascalsnail
Denote by P the Carath`eodory class of functions i.e. P = { p : p ( z ) = 1 + p z + p z + · · · , (cid:60) p ( z ) > z ∈ D ) } . The fundamental importance of P in geometric functions theoryrelies on the construction of several related families of analytic functions and is well known.Hence, various subclasses of P were defined and studied. Classical cases are related to thehalfplane and angular domain i.e. P ( α ) that denotes a subclass of P consisting with functionswith real part greater than α (0 ≤ α < P γ the class with argument between − γπ/ γπ/ < γ ≤ P were determined by the fact that somefunctionals are contained in convex subdomains of right halfplane. Therefore any subfamilyof halfplane domains were considered in the context to a subfamily of P . Hence a definitionof the domains related to the Pascal snail was a motivation to the definition of some subclassof P associated with such domains. To do this we first translate a domain L α,β,γ ( D ) witha vector (1 ,
0) in order to obtain a domain D α,β,γ contained in a right halfplane such that1 ∈ D α,β,γ . The boundary of the domain D α,β,γ is described as follows: ∂ D α,β,γ = (cid:40) u + iv : (cid:0) (2 − γ )( u −
1) + ( α + β )(( u − + v ) (cid:1) (1 + αβ ) + 4(1 − γ ) v (1 − αβ ) − (( u − + v ) = 0 (cid:41) . We note that D α,β,γ is contained in a halfplane (cid:60) w > L , where L is given in Corollary1.2. Anyway, there is substantial difference between D α,β,γ and a halfplane because D α,β,γ is not always a convex domain. However, when α = 0 and β → − then D α,β,γ tends to a NALYTIC REPRESENTATION OF THE GENERALIZED PASCAL SNAIL AND ITS APPLICATIONS 9 halfplane (cid:60) w > γ −
1. Thus D α,β,γ provides a natural bridge between the convex and theconcave domains.Now, we define a function T α,β,γ as(2.1) T α,β,γ ( z ) = 1 + L α,β,γ ( z ) , that map D univalently onto a domain D α,β,γ . Rewriting Corollary 1.2 for the function T α,β,γ we conclude the following theorem. Theorem 2.1.
Let − < α ≤ β < , ≤ γ < , and let T α,β,γ ( · ) be defined by (2.1) . Then (cid:60) {T α,β,γ ( z ) } > L ( α, β, γ ) and (cid:60) {T α,β,γ ( z ) } < M ( α, β, γ ) , where L ( α, β, γ ) and M ( α, β, γ ) are given in Corollary 1.2. Now, we are ready to construct a class P snail ( α, β, γ ) as follows P snail ( α, β, γ ) = { p ∈ P : p ( D ) ⊂ D α,β,γ } = { p ∈ P : p ≺ T α,β,γ } . The classes ST snail ( α, β, γ ) , CV snail ( α, β, γ ) and their properties In this Section we give a concise presentation of some families of analytic functions relatedto the generalized Pascal snail T α,β,γ . We will study some subclasses of S with functionsanalytic and univalent in D of the form(3.1) f ( z ) = z + ∞ (cid:88) n =2 a n z n ( z ∈ D ) . We also recall a class ST ( β ) ⊂ S , called starlike functions of order ≤ β <
1, that consistof functions f satisfying a condition (cid:60) (cid:8) zf (cid:48) ( z ) /f ( z ) (cid:9) > β ( z ∈ D )and a class CV ( β ), called convex functions of order ≤ β <
1, with analytic condition (cid:60) (cid:8) zf (cid:48)(cid:48) ( z ) /f (cid:48) ( z ) (cid:9) > β ( z ∈ D ) . Let f and g be analytic in D . Then the function f is said to subordinate to g in D writtenby f ( z ) ≺ g ( z ), if there exists a self-map of the unit disk ω , analytic in D with ω (0) = 0 andsuch that f ( z ) = g ( ω ( z )). If g is univalent in D , then f ≺ g if and only if f (0) = g (0) and f ( D ) ⊂ g ( D ).Also, let ST [ β ] be the subclass of ST defined by ST [ β ] := (cid:26) f ∈ A : zf (cid:48) ( z ) f ( z ) ≺ − βz (cid:27) where − ≤ β ≤ , β (cid:54) = 0. Notice that for β = ± w = 1 / (1 − βz ) maps theunit disc D onto the half-plane (cid:60) w > /
2, and for − < β < w = 1 / (1 − βz )maps the unit disc D onto the disc D ( C ( β ) , R ( β )) with the center C ( β ) = 1 / (1 − β ) and theradius R ( β ) = | β | / (1 − β ). Lemma 3.1.
Let − < α ≤ β < , ≤ γ < , and L α,β,γ be defined by (1.1) . Then L α,β,γ is starlike in D , moreover L α,β,γ ( z )2 − γ ∈ ST (cid:18) − | αβ | (1 + | α | ) (1 + | β | ) (cid:19) and L α,β,γ ( z )2 − γ ∈ CV ( t ( α, β )) , where (3.2) 0 ≤ t ( α, β ) = − | α | | α | + 1 − | β | | β | − αβ − αβ for αβ ≥ , − | α | | α | + 1 − | β | | β | − − αβ αβ for αβ < . Also, if | z | = r < , then (see Fig. 1, Fig. 2 and Fig. 6) max | z | = r | L α,β,γ ( z ) | = L α,β,γ ( r ) for αβ > with α + β > or α = 0 , − L α,β,γ ( − r ) for αβ > with α + β < or β = 0 , L α,β,γ ( r ) for αβ < with α + β > , − L α,β,γ ( − r ) for αβ < with α + β < , | L α, − α,γ ( ± r ) | for α + β = 0 , min | z | = r | L α,β,γ ( z ) | = − L α,β,γ ( − r ) for αβ > with α + β > or α = 0 , L α,β,γ ( r ) for αβ > with α + β < or β = 0 , − γ ) r √ | αβ | ( β − α )(1 − αβr ) for αβ < with α + β > , − γ ) r √ | αβ | ( β − α )(1 − αβr ) for αβ < with α + β < , | L α, − α,γ ( ± ir ) | for α + β = 0 . Proof.
A straightforward calculation shows that G := L α,β,γ satisfy (cid:60) (cid:26) zG (cid:48) ( z ) G ( z ) (cid:27) = 1 + (cid:60) (cid:26) αz − αz (cid:27) + (cid:60) (cid:26) βz − βz (cid:27) > − | α | | α | − | β | | β | , from which the result concerning starlikeness follows.In addition, we have1 + zG (cid:48)(cid:48) ( z ) G (cid:48) ( z ) = 1 + αz − αz + 1 + βz − βz − αβz − αβz ( z ∈ D ) . Thus for θ ∈ [0 , π ) (cid:60) (cid:26) e iθ G (cid:48)(cid:48) ( e iθ ) G (cid:48) ( e iθ ) (cid:27) = 1 − α α − α cos θ + 1 − β β − β cos θ − − α β α β − αβ cos 2 θ . A convexity result yield from the estimating the value of the function g ( t ) of the variable t := cos θ of the form g ( t ) := 1 − α α − αt + 1 − β β − βt − − α β (1 + αβ ) − αβt , where − ≤ t ≤ θ ∈ [0 , π ) the function Q ( θ ) := (cid:12)(cid:12)(cid:12) L α,β,γ (cid:16) re iθ (cid:17)(cid:12)(cid:12)(cid:12) = 4(1 − γ ) r (1 + α r − α r cos θ ) (1 + β r − β r cos θ ) (0 < r < . We see that min or max of Q ( θ ) are attained at the critical points of the above function,equivalently 8(1 − γ ) r sin θ (cid:0) αβr cos θ − ( α + β )(1 + αβr ) (cid:1) = 0 . For αβ = 0 and α + β (cid:54) = 0 the only ones critical points are θ = 0 , θ = π . Next, let αβ > Q (cid:48) ( θ ) = 0 for θ = 0 and θ = π since | ( α + β )(1 + αβr ) / αβ | ≤ α + β >
0, then for such θ we have − L α,β,γ ( − r ) ≤ (cid:12)(cid:12)(cid:12) L α,β,γ (cid:16) re iθ (cid:17)(cid:12)(cid:12)(cid:12) ≤ L α,β,γ ( r ) . NALYTIC REPRESENTATION OF THE GENERALIZED PASCAL SNAIL AND ITS APPLICATIONS 11
And, if α + β <
0, then for we obtain L α,β,γ ( r ) ≤ (cid:12)(cid:12)(cid:12) L α,β,γ (cid:16) re iθ (cid:17)(cid:12)(cid:12)(cid:12) ≤ − L α,β,γ ( − r ) . For the case αβ <
0, the critical points are θ = 0 , θ = π , and the solutions of the equation(3.3) 4 αβr cos θ − ( α + β )(1 + αβr ) = 0 . We consider three separate cases, the first is α + β >
0. Then for critical points θ = 0 andsolutions of the equation (3.3). In this case we have4(1 − γ ) r (cid:112) | αβ | ( β − α )(1 − αβr ) ≤ (cid:12)(cid:12)(cid:12) L α,β,γ (cid:16) re iθ (cid:17)(cid:12)(cid:12)(cid:12) ≤ L α,β,γ ( r ) . The second case is α + β <
0. Then for critical points θ = π and solutions of the equation(3.3), we obtain 4(1 − γ ) r (cid:112) | αβ | ( β − α )(1 − αβr ) ≤ (cid:12)(cid:12)(cid:12) L α,β,γ (cid:16) re iθ (cid:17)(cid:12)(cid:12)(cid:12) ≤ − L α,β,γ ( − r ) . Finally for the case α + β = 0, the critical points are θ = 0 , θ = π/ , θ = π, θ = 3 π/ θ = 2 π . For such θ we conclude2(1 − γ ) r α r = | L α, − α,γ ( ± ir ) | ≤ (cid:12)(cid:12)(cid:12) L α, − α,γ (cid:16) re iθ (cid:17)(cid:12)(cid:12)(cid:12) ≤ | L α, − α,γ ( ± r ) | = 2(1 − γ ) r − α r . (cid:3) Let α = ± β . Form (3.2), the function L α,α,γ ( z ) / (2 − γ ) is univalent in D if t ( α, α ) = α − | α | − α ≥ − √ ≤ α ≤ − √ T α,β,γ and present variousrelations of that family with the previously known classes. Definition 3.2.
For − < α ≤ β <
1, and 0 ≤ γ < γ ≥ T ( α, β ) and T ( α, β ) definedby(3.4) T ( α, β ) = (cid:40) − (1 − αβ )[2 √ αβ (1 − α )(1 − β )+( α + β )(1 − αβ )](1+ αβ ) for ( α, β ) ∈ B , − (1+ α )(1+ β )2 otherwise, let ST snail ( α, β, γ ) denote the subfamily of S consisting of the functions f , satisfying thecondition(3.5) zf (cid:48) ( z ) /f ( z ) ≺ T α,β,γ ( z ) ( z ∈ D ) , and let CV snail ( α, β, γ ) be a class of analytic functions f such that(3.6) 1 + zf (cid:48)(cid:48) ( z ) /f (cid:48) ( z ) ≺ T α,β,γ ( z ) ( z ∈ D ) , where T α,β,γ is given by (2.1). Geometrically, the condition (3.5) and (3.6) means that theexpression zf (cid:48) ( z ) /f ( z ) or 1+ zf (cid:48)(cid:48) ( z ) /f (cid:48) ( z ) lies in a domain bounded by the generalized Pascalsnail T α,β,γ (Fig. 8) given by (cid:2) ( u − + v − a ( u − (cid:3) = c ( u − + d v , where a, c and d given by (1.14).By the properties of T α,β,γ , given in Theorem 2.1 we have(3.7) (cid:60) (cid:8) zf (cid:48) ( z ) /f ( z ) (cid:9) > L ( z ∈ D ) , for f ∈ ST snail ( α, β, γ ), and for f ∈ CV snail ( α, β, γ )(3.8) (cid:60) (cid:8) zf (cid:48)(cid:48) ( z ) /f (cid:48) ( z ) (cid:9) > L ( z ∈ D ) , where L = L ( α, β, γ ) is given in Corollary 1.2.Additionally CV snail ( α, β, γ ) ⊂ G for γ satisfying(3.9) γ ≥ γ ( α, β ) = (cid:40) − − αβ )[2 √ αβ (1 − α )(1 − β )+( α + β )(1 − αβ )]2(1+ αβ ) for ( α, β ) ∈ B , − α )(1+ β )4 otherwise, where G is the family of function univalent, convex in one direction, and satisfying (cid:60){ zf (cid:48)(cid:48) ( z ) /f (cid:48) ( z ) } > − /
2, see [12].Taking into account (3.2) the function L α,α,γ ( z ) / (2 − γ ) ∈ G for | α | ≤ − √
13, and L α, − α,γ ( z ) / (2 − γ ) ∈ G for − (cid:112) − √ ≤ α <
0, [8].Summarizing, T α,β,γ is a analytic univalent function with positive real part in D , T α,β,γ ( D )is symmetric with respect to the real axis, starlike with respect to T α,β,γ (0) = 1 and convexin one direction under some conditions on α and β . Moreover T (cid:48) α,β,γ (0) = 2(1 − γ ) > T α,β,γ ( D ) satisfies Ma and Minda condition [9]. We refer to [2, 3, 6, 11] for a detaileddiscussion about similar subclasses of related to functions mapping the unit disk onto domainscontained in a right halfplane and starlike with respect to 1.For β = α with − < α < β = − α with − < α <
0, the quantities T ( α, β ) and γ ( α, β ) are the following T ( α, α ) = − α (cid:16) − α α (cid:17) for − √ ≤ α < , − α − α for − < α ≤ − √ , T ( α, − α ) = 1 + α γ ( α, α ) = − α (cid:16) − α α (cid:17) for − √ ≤ α < , − α − α for − < α ≤ − √ , γ ( α, − α ) = 1 + 3 α . For γ = 1 /
2, classes ST snail ( α, α, /
2) and CV snail ( α, α, /
2) are defined under the condition0 ≤ α ≤ α , where α = 0 . . . . is a root of equation 8 α (cid:0) − α (cid:1) = (cid:0) α (cid:1) . Wealso note that ST snail ( α, − α, /
2) and CV snail ( α, − α, /
2) of starlike and convex functionsof Ma-Minda type [9], can not be defined, because it should satisfy γ ≥ α that is α ≤ - uv (a) α = 1 −√ , γ = 2 −√ γ ≥T ( α, − α )) - uv (b) α = 1 − √ , γ = 1 . −√ γ < T ( α, − α )) Figure 8.
Image of D under T α, − α,γ ( z ) NALYTIC REPRESENTATION OF THE GENERALIZED PASCAL SNAIL AND ITS APPLICATIONS 13
Further properties of T α,β,γ yield: f ∈ ST snail ( α, β, γ ) = ⇒ φ ( z ) := (cid:90) z (cid:18) f ( t ) t (cid:19) − L dt ∈ CV . Indeed, by logarithmic differentiation of φ (cid:48) ( z ) = (cid:16) f ( z ) z (cid:17) − L we obtain1 + zφ (cid:48)(cid:48) ( z ) φ (cid:48) ( z ) = 1 − L (cid:18) zf (cid:48) ( z ) f ( z ) − (cid:19) = 1 + 1 L − L zf (cid:48) ( z ) f ( z ) ( z ∈ D ) . Since f ∈ ST snail ( α, β, γ ), we conclude that (cid:60) (cid:26) zφ (cid:48)(cid:48) ( z ) φ (cid:48) ( z ) (cid:27) = 1 + 1 L − L (cid:60) (cid:26) zf (cid:48) ( z ) f ( z ) (cid:27) > z ∈ D ) . The equivalence g ∈ ST snail ( α, β, γ ) if and only if zg (cid:48) ( z ) /g ( z ) ≺ T α,β,γ ( z ) allows to deter-mine the structural formula for functions in ST snail ( α, β, γ ). A function g is in the class ST snail ( α, β, γ ) if and only if there exists an analytic function p ≺ T α,β,γ , such that(3.10) g ( z ) = z exp (cid:18)(cid:90) z p ( t ) − t dt (cid:19) . The above integral representation provides many examples of functions of the class ST snail ( α, β, γ ).Let p ( z ) = T α,β,γ ( z n ) ∈ ST snail ( α, β, γ ) for n = 1 , , . . . . Then, for α (cid:54) = β, n ≥
1, the functionΨ α,β,γ,n ( z ) = z exp (cid:18)(cid:90) z − γ ) t n − (1 − αt n )(1 − βt n ) dt (cid:19) = z (cid:18) − βz n − αz n (cid:19) − γ ) n ( α − β ) (3.11) = z + 2(1 − γ ) n z n +1 + (1 − γ )[2(1 − γ ) + n ( α + β )] n z n +1 + · · · , is extremal for several problems in the class ST snail ( α, β, γ ). For n = 1 we have(3.12) Ψ α,β,γ ( z ) := Ψ α,β,γ, ( z ) = z (cid:18) − βz − αz (cid:19) − γ ) α − β , and for α = β (3.13) Ψ α,α,γ,n ( z ) = z exp (cid:18) − γ ) z n n (1 − αz n ) (cid:19) = z + 2(1 − γ ) z n +1 n (1 − αz n ) + · · · and(3.14) Ψ α,α,γ ( z ) := Ψ α,α,γ, ( z ) = z exp (cid:18) − γ ) z − αz (cid:19) . We note that Ψ α,β,γ,n ( D ) is sunflower’s domain (Fig. 9).Indeed, for α (cid:54) = β let G ( t ) = (cid:12)(cid:12) Ψ α,β,γ,n ( e it ) (cid:12)(cid:12) = (cid:18) β − β cos nt α − α cos nt (cid:19) p , where p = − γn ( α − β ) and n ≥
2. Since G (cid:48) ( t ) = (cid:18) β − β cos nt α − α cos nt (cid:19) p − pn ( β − α )(1 − αβ ) sin nt (1 + α − α cos nt ) , - - - - uv - - - - uv Figure 9. Ψ α,β,γ,n ( D ) for α = − . , β = 0 . , γ = 0 . , n = 5 , t = kπn , where k = 0 , , , ..., n − G ( t ) alternately attains its maximum and minimum, equal (cid:16) β α (cid:17) p and (cid:16) − β − α (cid:17) p , respectively. - tH ( t ) (a) α = − . , β = 0 . , γ =0 . , n = 5 with ( γ < T ( α, β )) - tH ( t ) (b) α = 0 . , β = 0 . , γ =0 . , n = 5 with ( γ ≥ T ( α, β )) Figure 10.
Graph of function H ( t ).Additionally, the argument of Ψ i.e. H ( t ) = Arg Ψ α,β,γ,n ( e it ) = t − p tan − ( β − α ) sin nt αβ − ( α + β ) cos nt , is also alternately increasing and decreasing (see Fig. 10) as t ∈ [0 , π ) and γ > T ( α, β ),where T ( α, β ) is defined by (3.4).In the case α = β the function G ( t ) has the form G ( t ) = exp (cid:60) (cid:18) − γ ) z n n (1 − αz n ) (cid:19) = exp (cid:18) − γ )(cos nt − α ) n (1 + α − α cos nt ) (cid:19) , NALYTIC REPRESENTATION OF THE GENERALIZED PASCAL SNAIL AND ITS APPLICATIONS 15 whose behavior is similar to the behavior of G ( t ) for α (cid:54) = β . The same situation holds for H ( t ) , α = β .If γ < T ( α, β ), the function Ψ α,β,γ,n is not starlike in a whole unit disk as well as notunivalent there (Fig. 11). - - - - - - uv - - - - - - uv Figure 11. Ψ α,β,γ,n ( D ) for α = − . , β = 0 . , γ = 0 . , n = 5 , γ < T ( α, β ).From Lemma 3.1 it can be seen that the smallest disk with center (1 ,
0) that contains T α,β,γ ( z )and the largest disk with center at (1 ,
0) contained in T α,β,γ ( z ) (see Fig 12) are, below. Proposition 3.3.
Let − < α ≤ β < , αβ (cid:54) = ± . Then (3.15) T α,β,γ ( D ) ⊃ (cid:110) w ∈ C : | w − | < − γ )(1+ α )(1+ β ) (cid:111) for αβ > with α + β < or β = 0 , (cid:110) w ∈ C : | w − | < − γ )(1 − α )(1 − β ) (cid:111) for αβ > with α + β > or α = 0 , (cid:26) w ∈ C : | w − | < − γ ) √ | αβ | ( β − α )(1 − αβ ) (cid:27) for αβ < with α + β (cid:54) = 0 , (cid:110) w ∈ C : | w − | < − γ )1+ α (cid:111) for α + β = 0 , (3.16) T α,β,γ ( D ) ⊂ (cid:110) w ∈ C : | w − | < − γ )(1 − α )(1 − β ) (cid:111) for αβ (cid:54) = 0 with α + β > or α = 0 , (cid:110) w ∈ C : | w − | < − γ )(1+ α )(1+ β ) (cid:111) for αβ (cid:54) = 0 with α + β < or β = 0 , (cid:110) w ∈ C : | w − | < − γ )1 − α (cid:111) for α + β = 0 . The function Ψ α,β,γ given by (3.11) , and (3.14) shows that the bounds are the best possible. Theorem 3.4.
Let − < α ≤ β < , and let f be analytic in D . If P f = f /f (cid:48) ∈ST snail ( α, β, γ ) , then zf (cid:48) ( z ) f ( z ) ≺ z Ψ α,β,γ ( z ) ( z ∈ D ) . Proof.
Let p ( z ) = zf (cid:48) ( z ) /f ( z ). Then P f ( z ) = z/p ( z ) and zP (cid:48) f /P f = 1 − zp (cid:48) /p . Since P f ∈ ST snail ( α, β, γ ), we have − zp (cid:48) ( z ) p ( z ) ≺ T α,β,γ ( z ) − L α,β,γ ( z ) ( z ∈ D ) . ( - γ )( - α )( - β ) z + ( - γ ) r αβ ( β - α )( - αβ ) z + ( - γ ) z ( - α z )( - β z ) - γ + α + β z + - - uv Figure 12.
The range of the functions L α,β,γ , − γ )(1+ α )(1+ β ) z +1, − γ )(1 − α )(1 − β ) z +1and − γ ) √ | αβ | ( β − α )(1 − αβ ) z + 1 for α = − . γ = 0 .
93 and β = 0 . F defined by F ( z ) = (cid:90) z L α,β,γ ( t ) t dt = log (cid:18) Ψ α,β,γ ( z ) z (cid:19) where Ψ α,β,γ given by (3.12), is analytic in D , F (0) = F (cid:48) (0) − zF (cid:48)(cid:48) ( z ) F (cid:48) ( z ) = z L (cid:48) α,β,γ ( t ) L α,β,γ ( t ) ( z ∈ D ) . Taking into account Lemma 3.1, we deduce that the function F is convex in D . Applying[14], we conclude that − log p ( z ) ≺ log (cid:18) Ψ α,β,γ ( z ) z (cid:19) or log p ( z ) ≺ log (cid:18) z Ψ α,β,γ ( z ) (cid:19) , and by (3.12), the required result follows. (cid:3) Since for z ∈ D and α (cid:54) = β (cid:60) (cid:26) z Ψ α,β,γ ( z ) (cid:27) = (cid:12)(cid:12)(cid:12)(cid:12) − αz − βz (cid:12)(cid:12)(cid:12)(cid:12) − − γ ) β − α cos (cid:18) − γ ) β − α arg 1 − αz − βz (cid:19) and arg − αz − βz ∈ ( − π/ , π/
2) the above and Theorem 3.4 leads to the following conclusion.
Corollary 3.5.
Let f ∈ A be a locally univalent function. If P f = f /f (cid:48) ∈ ST snail ( α, β, γ ) with α (cid:54) = β and γ ≥ − β − α , then f ∈ ST . Now we get a representation of functions in class ST snail ( α, β, γ ) with the help of the class ST [ β ]. Lemma 3.6.
Let f ∈ ST snail ( α, β, γ ) with α, β (cid:54) = 0 and α (cid:54) = β . Then there exists h ∈ ST [ β ] ,and g ∈ ST [ α ] such that f ( z ) = z (cid:18) h ( z ) g ( z ) (cid:19) − γ ) β − α ( z ∈ D ) . NALYTIC REPRESENTATION OF THE GENERALIZED PASCAL SNAIL AND ITS APPLICATIONS 17
Proof.
Let f ∈ ST snail ( α, β, γ ). Then, by (3.10), there exists a self-map ω , which is analyticin D , ω (0) = 0 , | ω ( z ) | <
1, and such that f ( z ) = z exp (cid:18)(cid:90) z T α,β,γ ( ω ( t )) − t dt (cid:19) = z exp (cid:90) z q (cid:20) βω ( t ) t (1 − βω ( t )) − αω ( t ) t (1 − αω ( t )) (cid:21) dt = z z exp (cid:90) z βω ( t ) t [1 − βω ( t )] dtz exp (cid:90) z αω ( t ) t [1 − αω ( t )] dt q = z z exp (cid:90) z − βω ( t ) − t dtz exp (cid:90) z − αω ( t ) − t dt q = z (cid:18) h ( z ) g ( z ) (cid:19) q , where q = − γ ) β − α . The assertion now follows. (cid:3) From the relation h ∈ CV snail ( α, β, γ ) if and only if 1 + zh (cid:48)(cid:48) ( z ) /h (cid:48) ( z ) ≺ T α,β,γ ( z ) weobtain the structural formula for functions in CV snail ( α, β, γ ). A function h is in the class CV snail ( α, β, γ ) if and only if there exists an analytic function p with p ≺ T α,β,γ , such that(3.17) h ( z ) = (cid:90) z exp (cid:18)(cid:90) w p ( t ) − t dt (cid:19) dw. The above representation supply many examples of functions in class CV snail ( α, β, γ ). Let p ( z ) = T α,β,γ ( z ) ∈ CV snail ( α, β, γ ), then for some n ≥ α (cid:54) = β , the functions(3.18) K α,β,γ,n ( z ) = (cid:90) z exp (cid:18)(cid:90) w − γ ) t n − (1 − αt n ) (1 − βt n ) dt (cid:19) dw = (cid:90) z (cid:18) − αt n − βt n (cid:19) − γ ) n ( α − β ) dt, are extremal functions for several problems in the class CV snail ( α, β, γ ). For n = 1 we have(3.19) K α,β,γ ( z ) := K α,β,γ, ( z ) = (cid:90) z (cid:18) − αt − βt (cid:19) − γ ) α − β dt. and for α = β (3.20) K α,α,γ,n ( z ) = (cid:90) z exp (cid:18) − γ ) t n n (1 − αt n ) (cid:19) dt and K α,α,γ ( z ) := K α,α,γ, ( z ) . Now we get a representation of functions in class CV snail ( α, β, γ ) with the help of class ST [ β ]. From Lemma 3.6, we conclude the following Corollary. Corollary 3.7.
Let f ∈ CV snail ( α, β, γ ) with α, β (cid:54) = 0 and α (cid:54) = β . Then there exists h ∈ST [ β ] and g ∈ ST [ α ] such that f (cid:48) ( z ) = (cid:18) h ( z ) g ( z ) (cid:19) − γ ) β − α ( z ∈ D ) . From (3.15), we conclude that f ∈ ST snail ( α, β, γ ) if and only if (cid:12)(cid:12)(cid:12)(cid:12) zf (cid:48) ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) < L = − γ )(1+ α )(1+ β ) for αβ > with α + β < or β = 0 , − γ )(1 − α )(1 − β ) for αβ > with α + β > or α = 0 , − γ ) √ | αβ | ( β − α )(1 − αβ ) for αβ < with α + β (cid:54) = 0 , − γ )1+ α for α + β = 0and the fact that f ∈ CV snail ( α, β, γ ) if and only if zf (cid:48) ( z ) ∈ ST snail ( α, β, γ ), we get thefollowing conclusions. Proposition 3.8.
Let − < α ≤ β < . The classes ST snail ( α, β, γ ) and CV snail ( α, β, γ ) are nonempty. The following functions are the examples of their members. (1) Let a n ∈ C with n = 2 , , . . . . Then f ( z ) = z + a n z n ∈ ST snail ( α, β, γ ) ⇐⇒ | a n | ≤ Ln − L . (2) Let a n ∈ C with n = 2 , , . . . . Then f ( z ) = z + a n z n ∈ CV snail ( α, β, γ ) ⇐⇒ n | a n | ≤ Ln − L . (3) Let A ∈ C . Then z/ (1 − Az ) ∈ ST snail ( α, β, γ ) ⇐⇒ | A | ≤ L . (4) Let A ∈ C . Then z/ (1 − Az ) ∈ CV snail ( α, β, γ ) ⇐⇒ | A | ≤ L . (5) Let A ∈ C . Then z exp( Az ) ∈ ST snail ( α, β, γ ) ⇐⇒ | A | ≤ L. (6) Let A ∈ C . Then exp( Az ) − A ∈ CV snail ( α, β, γ ) ⇐⇒ < | A | ≤ L, where L is given in the Corollary 3.7.The following corollary is the consequence of Lemma 3.1, and Theorems in [9]. Corollary 3.9.
For − < α ≤ β < , | z | = r < , and f ∈ ST snail ( α, β, γ ) , it holds − Ψ α,β,γ ( − r ) ≤ | f ( z ) | ≤ Ψ α,β,γ ( r ) , Ψ (cid:48) α,β,γ ( − r ) ≤ (cid:12)(cid:12) f (cid:48) ( z ) (cid:12)(cid:12) ≤ Ψ (cid:48) α,β,γ ( r ) for αβ > with α + β > or α = 0 , Ψ (cid:48) α,β,γ ( r ) ≤ (cid:12)(cid:12) f (cid:48) ( z ) (cid:12)(cid:12) ≤ Ψ (cid:48) α,β,γ ( − r ) for αβ > with α + β < or β = 0 , | Arg { f ( z ) /z }| ≤ max | z | = r Arg { Ψ α,β,γ ( z ) /z } . Equalities in the above inequalities hold at a given point other than origin for the functions (3.21) ψ α,γ,µ ( z ) = µ Ψ α,β,γ ( µz ) ( | µ | = 1) . Moreover (3.22) f ( z ) z ≺ Ψ α,β,γ ( z ) z ( z ∈ D ) . If f ∈ ST snail ( α, β, γ ) , then either f is a rotation of Ψ α,β,γ given by (3.12) and (3.14) or { w ∈ C : | w | ≤ − Ψ α,β,γ ( − } ⊂ f ( D ) , where − Ψ α,β,γ ( −
1) = lim r → − [ − Ψ α,β,γ ( − r )] . Corollary 3.10.
Let − < α ≤ β < . If f ∈ CV snail ( α, β, γ ) and | z | = r < , then − K α,β,γ ( − r ) ≤ | f ( z ) | ≤ K α,β,γ ( r ) ,K (cid:48) α,β,γ ( − r ) ≤ | f (cid:48) ( z ) | ≤ K (cid:48) α,β,γ ( r ) , | Arg (cid:8) f (cid:48) ( z ) (cid:9) | ≤ max | z | = r Arg (cid:8) K (cid:48) α,β,γ ( z ) (cid:9) . Equalities in the above inequalities hold at a given point other than for functions µK α,β,γ ( µz ) with ( | µ | = 1) . Moreover f (cid:48) ( z ) ≺ K (cid:48) α,β,γ ( z ) ( z ∈ D ) . If f ∈ CV snail ( α, β, γ ) , then either f is a rotation of K α,β,γ given by (3.19) and (3.20) or { w ∈ C : | w | ≤ − K α,β,γ ( − } ⊂ f ( D ) , where − K α,β,γ ( −
1) = lim r → − [ − K α,β,γ ( − r )] . Theorem 3.11.
Let − < α < β < . If f ∈ ST snail ( α, β, γ ) , then (1) (cid:60) (cid:26) f ( z ) z (cid:27) > (cid:18) α β (cid:19) − γβ − α for T ( α, β ) ≤ − β − α ≤ γ ( z ∈ D ) , NALYTIC REPRESENTATION OF THE GENERALIZED PASCAL SNAIL AND ITS APPLICATIONS 19 (2) (cid:60) (cid:26) f ( z ) z (cid:27) β − α − γ > α β ( z ∈ D ) , (3) (cid:12)(cid:12)(cid:12)(cid:12) Arg (cid:26) f ( z ) z (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − γ ) β − α sin − (cid:18) | z | ( β − α )1 − | z | αβ (cid:19) ( z ∈ D ) . Proof.
Let q := − γ ) β − α .Case 1. From 1 − β − α ≤ γ it follows that 0 < (2 − γ ) / ( β − α ) ≤
1, and from f ∈ST snail ( α, β, γ ) it follows that T ≤ − β − α . Then, making use Corollary 3.9 and Lemma1.3, we conclude that (cid:60) (cid:26) f ( z ) z (cid:27) > (cid:60) (cid:26) Ψ α,β,γ ( z ) z (cid:27) = (cid:60) (cid:26)(cid:18) − αz − βz (cid:19) q (cid:27) ≥ (cid:26) (cid:60) (cid:18) − αz − βz (cid:19)(cid:27) q > (cid:18) α β (cid:19) q . The function ψ α,γ,µ given by (3.21), shows that the bound is the best possible.Case 2. From Corollary 3.9 we have (cid:20) f ( z ) z (cid:21) /q ≺ (cid:20) Ψ α,β,γ ( z ) z (cid:21) /q . Thus (cid:60) (cid:26) f ( z ) z (cid:27) /q > (cid:60) (cid:26) − αz − βz (cid:27) > α β . Case 3. By Corollary 3.9 it is enough to consider Arg { Ψ α,β,γ ( z ) /z } . Since the image of thedisk { z ∈ C : | z | ≤ r } by the function w = Ψ α,β,γ ( z ) /z or w /q = (1 − αz ) / (1 − βz ) is con-tained in closed disc with center (cid:0) − αβr (cid:1) / (cid:0) − β r (cid:1) and radius ( r ( β − α )) / (cid:0) − β r (cid:1) .Therefore (cid:12)(cid:12)(cid:12)(cid:12) w /q − − αβr − β r (cid:12)(cid:12)(cid:12)(cid:12) ≤ r ( β − α )1 − β r and (cid:12)(cid:12)(cid:12) Arg w /q (cid:12)(cid:12)(cid:12) < π . Thus (cid:12)(cid:12)(cid:12)
Arg w /q (cid:12)(cid:12)(cid:12) ≤ sin − (cid:18) r ( β − α )1 − r αβ (cid:19) . The proof is now complete. (cid:3)
It is clear that f ( z ) ∈ CV snail ( α, β, γ ) if and only if zf (cid:48) ( z ) ∈ ST snail ( α, β, γ ). Using thesame notation and the same reasoning as in the proof of Theorem 3.11 we have the followingCorollary. Corollary 3.12.
Let − < α < β < . If f ∈ CV snail ( α, β, γ ) , then (1) (cid:60) { f (cid:48) ( z ) } > (cid:18) α β (cid:19) − γβ − α for T ( α, β ) ≤ − β − α ≤ γ ( z ∈ D ) , (2) (cid:60) { f (cid:48) ( z ) } β − α − γ > α β ( z ∈ D ) , (3) | Arg { f (cid:48) ( z ) }| ≤ − γ ) β − α sin − (cid:18) | z | ( β − α )1 − | z | αβ (cid:19) ( z ∈ D ) . Condition T ( α, β ) ≤ − β − α ≤ γ in Theorem 3.11 for requirement β = − α are equivalentto conditions 1 − √ ≤ α < γ ≥ α , and so we have the following. Corollary 3.13.
For − √ ≤ α < and γ ≥ α , we have: f ∈ ST snail ( α, − α, γ ) = ⇒ (cid:60) (cid:26) f ( z ) z (cid:27) > (cid:18) − α α (cid:19) − γα ( z ∈ D ) , and f ∈ CV snail ( α, − α, γ ) = ⇒ (cid:60) (cid:8) f (cid:48) ( z ) (cid:9) > (cid:18) − α α (cid:19) − γα ( z ∈ D ) . Acknowledgments
The authors thank the editor and the anonymous referees for constructive and pertinentsuggestions.
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Competing Interests
The authors declare that they have no competing interests.
Funding
This work was partially supported by the Center for Innovation and Transfer of Natural Sci-ences and Engineering Knowledge, Faculty of Mathematics and Natural Sciences, Universityof Rzeszow.
Authors’ Contributions
Each of the authors contributed to each part of this study equally, all authors read andapproved the final manuscript.
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