Uniformly Convex and Uniformly Starlike Functions

Theorem 2.1. [16, Theorem 1, p. 365] The function f is in UST if and only if (2.1) Re (z −ζ)f ′(z) f(z) −f(ζ)  ≥0, z, ζ ∈D. By taking ζ = −z in the above theorem, evidently the class UST ⊂S∗ s and hence |an| ≤1 for f ∈UST . A better bound |an| ≤2/n for f ∈UST , proved by Charles Horowitz, was also reported in Goodman [16, Theorem 4, p. 368]. The proof involved showing UST is a subclass

Theorem 2.2. [16, Lemma 1, p. 365] Let f ∈UST , and define p0, p1, q0, q1 by p0(ζ) = f(ζ) ζ , p1(z) = f(ζ)(1 −2a2ζ) −ζ ζ2 , q0(ζ) = f(z) zf ′(z), q1(z) = f(z) −z z2f ′(z) . Then |p1(ζ)| ≤2 Re(p0(ζ)), and

Theorem 2.2 and the coefficient estimate |an| ≤2/n for f ∈UST yield the growth inequality for UST : r 1 + 2r ≤|f(z)| ≤−r + 2 ln 1 1 −r , |z| = r < 1. This inequality provides the lower bound for the Koebe constant for the family UST : 1 3 ≤K(UST ) ≤1 − √ 3 4 . The upper bound follows from the function f given by

Theorem 2.3 ([32, Theorem 1, p. 450]). Let f ∈A. Then f ∈UST if and only if for all complex numbers α, β with |α| < 1 and |β| < 1, Re

Theorem 2.4. [51, Lemma 3.3, p. 236] The function f ∈UST if and only if (2.2) Re  f(z) −f(xz) (1 −x)zf ′(z)  ≥0, z ∈D, |x| = 1. Let G denote the subset of A having the property P. If, for every f ∈F, r−1f(rz) ∈G for r ≤R, and R is the largest number for which this holds, then R is the G-radius (or the radius of the property P) in F. Thus, the radius of a property P in the set F is the largest number R such that every function in the set F has the

Theorem 2.5. (1) The UST -radius for the class of univalent func- tions S is r0 ≈0.3691. (2) The UST -radius r∗ 0 for the class S∗satisfies 0.369 < r∗ 0 ≤1/ √ 7. (3) The UST -radius for the class of convex func- tions C is 1/ √ 2. (4) The UST -radius for the class of pre-starlike functions is at least (1 + α)/(1 −α) for

Theorem 3.1 ([15, Theorem 1, p. 88]). The function f belongs to UCV if and only if (3.1) Re  1 + (z −ζ)f ′′(z) f ′(z)  ≥0, z, ζ ∈D. Though the class C is a linear invariant family, the class UCV is not. This was proved by Goodman [15,

Theorem 5, p. 90] by using the function F(z) = z 1 −Az . This function F ∈UCV if and only if |A| ≤1/3. From the geometric definition or from Theorem 3.1, it is evident that UCV ⊂CV. However, by taking ζ = −z

Theorem 3.2. Let f ∈A. Then f ∈UCV if and only if (3.2) Re  1 + zf ′′(z) f ′(z)  >

Theorem 3.3. Let f be a function of the form f(z) = z −P∞ n=2 anzn with an ≥0. Then f ∈UCV ⇔ ∞ X n=2 n(2n −1)an ≤1 and f ∈SP ⇔ ∞ X n=2 (2n −1)an ≤1. Denote the class of all functions with negative coef-

Theorem 3.3, it can be seen [50] that f(z) = z −Anzn ∈SP ⇔|An| ≤ 1 2n −1, and f ∈UCV ⇔|An| ≤ 1 n(2n −1).

Theorem 3.4. [30] If f ∈C(ϕ), then, for |z| = r, k′ ϕ(−r) ≤|f ′(z)| ≤k′ ϕ(r), −kϕ(−r) ≤|f(z)| ≤kϕ(r),

Theorem 3.5 (Ruscheweyh [55, Theorem 1, p. 275]). Let φ be a convex function defined in D with φ(0) = 1. Define F by F(z) = z exp Z z 0 φ(x) −1 x dx  . The function f belongs to S∗(φ) if and only if for all |s| ≤1 and |t| ≤1, sf(tz) tf(sz) ≺sF(tz)

Theorem 3.6. [50, Theorem 5, p. 194] Let f ∈SP and f(z) = z + P∞ n=2 anzn. Then (3.6) |a2| ≤c, and |an| ≤ c n −1 n Y k=3  1 + c

Theorem 3.7. Let φ(z) = 1 + B1z + B2z2 + · · · . If f(z) = z + a2z2 + a3z3 + · · · ∈C(ϕ), then |a3 −µa2 2| ≤          

Theorem 3.8. [58, Theorem 2.4, p. 54] Let α ≤1, f ∈Rα and g ∈S∗(α). Then, for any analytic function H ∈H(D), f ∗Hg f ∗g (D) ⊂co(H(D)), where co(H(D)) denote the closed convex hull of H(U).

Theorem 3.9. [49, Theorem 3.6, p. 131] Let ϕ be a convex function with Re ϕ(z) ≥α, α < 1. If f ∈Rα and g ∈S∗(ϕ), then f ∗g ∈S∗(ϕ). The proof of this theorem follows readily from The- orem 3.8 by putting H(z) = zg′(z)/g(z). In view of the fact that f ∈C(ϕ) if and only if zf ′ ∈S∗(ϕ), an immediate consequence of the above theorem is the cor- responding result for C(ϕ): if f ∈Rα and g ∈C(ϕ), then f ∗g ∈C(ϕ) for any convex function ϕ with Re ϕ(z) ≥α. In particular, the classes UCV and SP are closed

Corollary 3.1. [40] Let Γ1(f(z)) = zf ′(z), Γ2(f(z)) = 1 2[f(z) + zf ′(z)] Γ3(f(z)) = k + 1 zk Z z 0 ζk−1f(ζ)dζ, Re k > 0 Γ4(f(z)) = Z z 0 f(ζ) −f(ηζ) ζ −ηζ dζ, |η| ≤1, η ̸= 1.

Theorem 3.10. [27, Theorem 1, p. 768] Let a, b ∈C − {0} and c > |a| + |b| + 2. If Γ(c −|a| −|b|)Γ(c) Γ(c −|a|)Γ(c −|b|)×  1 + 2(|a|)2(|b|)2 (c −2 −|a| −|b|)2 + 5|ab| c −|a| −|b| −1  ≤2, then zF(a, b; c; z) ∈UCV. They also obtained a weaker condition on the pa-

Theorem 3.11. [27, Theorem 4, p. 771] Let a, b ∈C − {0} and c > |a| + |b| + 1. If 2(1 −β) cos η

Theorem 3.12. [59, Theorem 1, p. 320] Let fi ∈UCV and αi’s be real numbers such that αi ≥0, and Pn 1 αi ≤ 1. Then the function g(z) = Z z 0 n Y i=1 [f ′ i(ζ)]αidζ belongs to UCV. As an immediate consequence of this theorem, the function g defined by

Theorem 3.13. [59, Theorem 2, p. 320] If f ∈A sat- isfies

Theorem 3.14 ([22]). Let f ∈S. Then the following are equivalent: (1) f ∈k −UCV, (2) the inequality Re  1 + (z −ζ)f ′′(z) f ′(z)  ≥0 holds for all z ∈D and for all |ζ| ≤k, (3) the inequality Re  1 + zf ′′(z)

Theorem 3.15. [46] Let |α| < π 2 . A function f ∈A belongs to USP(α) if and only if Re  e−iα (z −ζ)f ′(z) f(z) −f(ζ)  ≥0, z ̸= ζ, z, ζ ∈D. The arc Γw is convex α-spirallike if arg z′′(t) z′(t) + z′(t)f ′′(z)

Theorem 3.16. [46] Let f ∈A. The the following are equivalent. (1) f ∈UCSP(α), (2) f satisfies the inequality Re  e−iα  1 + (z −ζ)f ′′(z) f ′(z)  ≥0, z ̸= ζ, z, ζ ∈D, (3) f satisfies the inequality

Theorem 3.17. The SP-radius of the class S of univa- lent functions is 0.33217 and the SP-radius of the class S∗of starlike functions is 1/3 ≈0.3333 [50, Corollary 3, Theorem 4, p. 192]. The SP-radius of the class C of