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Abstract
A normalized analytic function f defined on the open unit disk in the com-
plex plane is in the class SL if zf ′(z)/f(z) lies in the region bounded by the right-half of
the lemniscate of Bernoulli given by |w2 −1| < 1. In the present investigation, the SL-
radii for certain well-known classes of functions are obtained. Radius problems associated
with the left-half plane are also investigated for these classes.
Results & Lemmas (10)
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Lemma 1.1.
Lemma 1.1. [7] If p ∈Pn, then
Lemma 1.1. [7] If p ∈Pn, then
Lemma 1.2.
Lemma 1.2. [12] If p ∈Pn[A, B], then p(z) −1 −ABr2n 1 −B2r2n ≤(A −B)rn 1 −B2r2n (|z| = r < 1). In particular, if p ∈Pn(α), then p(z) −1 +…
Lemma 1.2. [12] If p ∈Pn[A, B], then p(z) −1 −ABr2n 1 −B2r2n ≤(A −B)rn 1 −B2r2n (|z| = r < 1). In particular, if p ∈Pn(α), then p(z) −1 + (1 −2α)r2n 1 −r2n ≤2(1 −α)rn 1 −r2n (|z| = r < 1). 2. The SLn-Radius Problems In this section, three special classes of functions will be considered. First is the class Sn :=
Theorem 2.1.
Theorem 2.1. The SLn-radius for the class Sn is RSLn(Sn) = √ 2 −1 n + q n2 + ( √ 2 −1)2
Theorem 2.1. The SLn-radius for the class Sn is RSLn(Sn) = √ 2 −1 n + q n2 + ( √ 2 −1)2
Lemma 2.2.
Lemma 2.2. For 0 < a < √ 2, let ra be given by ra = ( √ 1 −a2 −(1 −a2) 1/2 (0 < a ≤2 √ 2/3) √ 2 −a (2 √ 2/3 ≤a <
Lemma 2.2. For 0 < a < √ 2, let ra be given by ra = ( √ 1 −a2 −(1 −a2) 1/2 (0 < a ≤2 √ 2/3) √ 2 −a (2 √ 2/3 ≤a <
Theorem 2.3.
Theorem 2.3. The SLn-radius for the class CSn(α) is given by RSLn(CSn(α)) = √ 2 −1 (1 + n −α) + q (1 + n −α)2 + (1 −2α + √ 2)( √ 2 −1)…
Theorem 2.3. The SLn-radius for the class CSn(α) is given by RSLn(CSn(α)) = √ 2 −1 (1 + n −α) + q (1 + n −α)2 + (1 −2α + √ 2)( √ 2 −1)
Theorem 2.4.
Theorem 2.4. Let −1 < B < A ≤1 and either (i) 1 + A ≤ √ 2(1 + B) and 2 √ 2(1 − B2) ≤3(1 −AB) < 3 √ 2(1 −B2), or (ii) (A −B)(1 −B2) + (1…
Theorem 2.4. Let −1 < B < A ≤1 and either (i) 1 + A ≤ √ 2(1 + B) and 2 √ 2(1 − B2) ≤3(1 −AB) < 3 √ 2(1 −B2), or (ii) (A −B)(1 −B2) + (1 −B2)2 ≤(1 − B2) p (1 −B2) −(1 −AB)2+(1−AB)2 and 2 √ 2(1−B2) ≥3(1−AB). Then ST n[A, B] ⊂ SLn.
Theorem 2.5.
Theorem 2.5. Let −1 ≤B < A ≤1, with B ≤0. Then the SLn-radius for the class ST n[A, B] is RSLn (ST n[A, B]) = min 1, 2( √ 2 −1) (A…
Theorem 2.5. Let −1 ≤B < A ≤1, with B ≤0. Then the SLn-radius for the class ST n[A, B] is RSLn (ST n[A, B]) = min 1, 2( √ 2 −1) (A −B) + q (A −B)2 + 4( √
Theorem 2.6.
Theorem 2.6. Assume that f ∈ST n[A, B] and 0 < B < A ≤1. Let R1 be given by R1 =
Theorem 2.6. Assume that f ∈ST n[A, B] and 0 < B < A ≤1. Let R1 be given by R1 =
Theorem 3.1.
Theorem 3.1. The Mn(β)-radius of functions in Sn is given by RMn(β)(Sn) = " β −1 n + p n2 + (β −1)2 #1/n.
Theorem 3.1. The Mn(β)-radius of functions in Sn is given by RMn(β)(Sn) = " β −1 n + p n2 + (β −1)2 #1/n .
Theorem 3.2.
Theorem 3.2. The Mn(β)-radius of functions in CSn(α) is given by RMn(β)(CSn(α)) = β −1 (1 + n −α) + p (1 + n −α)2 + (β −1)(1 + β −2α).
Theorem 3.2. The Mn(β)-radius of functions in CSn(α) is given by RMn(β)(CSn(α)) = β −1 (1 + n −α) + p (1 + n −α)2 + (β −1)(1 + β −2α) .
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