Nehari's univalence criteria, pre-Schwarzian derivative and applications

Theorem 2.1. If f ∈N2(k), 0 ≤k ≤2, then (2) |Tf(z)| ≤2|z| −2β2Ak(|z|) 1 −|z|2 ,

Lemma 2.2. If f ∈M2(k), 0 ≤k ≤2, then |Tf(z)| ≤2|z| + √ 4 −2k 1 −|z|2 . Equality holds at a single z ̸= 0 if and only if f is a suitable rotation of F0(z), where F0(z) = e √ 4 −2k tanh−1(z) −1 √ 4 −2k = 1 + z 1 −z

Corollary 2.3. [2, Lemma 1] If f ∈N2(2) then |Tf(z)| ≤ 2|z| 1 −|z|2. Equality holds at a single z ̸= 0 if and only if f is a rotation of 1 2 ln 1 + z 1 −z. Similarly, the next result is stated as follows:

Theorem 2.4. If f ∈N0(k), 0 ≤k ≤π2/2, then |Tf(z)| ≤ √ 2k tan r k 2|z| ! . Equality holds at a single z ̸= 0 if and only if f is a rotation of F1(z), where F1(z) = r 2 k tan r

Corollary 2.5. If f ∈N0(π2/2) then |Tf(z)| ≤π tan π 2 |z|  . The equality holds at a single z ̸= 0 if and only if f is a rotation of 2 π tan π 2 z  . Next we present a similar result for functions in the class N1(k). Since we use the same technique and it involves solution of a differential equation, as a supplementary result we

Theorem 2.6. A solution of the differential equation w′(z) = 1 2w2(z) + k 1 −z2, 0 ≤k ≤4,

Theorem 2.7. If f ∈N1(k), 0 ≤k ≤4, then |Tf(z)| ≤k|z| F[(−1/4)(−3 + √ 1 + 2k), (1/4)(3 + √ 1 + 2k); 3/2; |z|2] F[(−1/4)(1 + √ 1 + 2k), (1/4)(−1 + √ 1 + 2k); 1/2; |z|2]  . Equality holds at a single z ̸= 0 if and only if f is a rotation of F2(z), where

Corollary 2.8. If f ∈N1(4) then |Tf(z)| ≤ 4|z| 1 −|z|2. Equality holds at a single z ̸= 0 if and only if f is a rotation of 1 4  2z 1 −z2 + ln 1 + z 1 −z  . 3. Schwarzian derivative and John domains This section is devoted to the study of functions in Nehari-type classes. We begin with

Lemma 3.1. [3, Lemma 2] Let f be analytic and univalent in D. Then f(D) is a John domain if and only if there exists 0 < x < 1 such that sup |ζ|=1 sup r<1 (1 −ρ2)|f ′(ρζ)| (1 −r2)|f ′(rζ)| < 1, ρ = x + r 1 + xr. This characterization plays an important role to prove the following necessary condition for bounded John domains f(D), when f ∈N2(2) (see [3, Theorem 4]). In fact, in its proof, a relationship between the derivatives Sf and Tf as well as an upper bound for |Tf| are used.

Lemma 3.2. [3, Theorem 4] Let f ∈N2(2) and f(D) be a bounded John domain. Then lim sup |z|→1 (1 −|z|2)Re (zTf(z)) < 2. Naturally, one can ask the analog of Lemma 3.2 for the family N2(k), 0 ≤k < 2. From [2, Lemma 1] it is clear that for all bounded mappings, lim sup |z|→1 (1 −|z|2)2|Sf(z)| ≤k =⇒lim sup |z|→1 (1 −|z|2)|Tf(z)| ≤k. From this, we conclude that, for f ∈N2(k), 0 ≤k ≤2, lim sup |z|→1 (1 −|z|2)Re (zTf(z)) ≤k.

Theorem 3.3. Let f ∈M2(k), 0 ≤k < 2. Then lim sup |z|→1 (1 −|z|2)Re (zTf(z)) < 2 + √ 4 −2k.