Ma-Minda φ-classes studied in this paper:
Results & Lemmas (16)
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Lemma 1.
Lemma 1. [12] If 𝑝∈𝒫is of the form 𝑝(𝑧) = 1 + 𝑐1𝑧+ 𝑐2𝑧2 +..., then |𝑐𝑛| ≤2 ∀𝑛∈N.
Lemma 1. [12] If 𝑝∈𝒫is of the form 𝑝(𝑧) = 1 + 𝑐1𝑧+ 𝑐2𝑧2 + . . . , then |𝑐𝑛| ≤2 ∀𝑛∈N.
Lemma 2. · coeff
Lemma 2. [9] For a Schwarz function 𝑤(𝑧) = 𝑐1𝑧+ 𝑐2𝑧2 +..., and for any 𝜇∈C we have ⃒⃒𝑐2 −𝜇𝑐2 1 ⃒⃒≤max 1, |𝜇|. 3. Coefficient estimates for…
Lemma 2. [9] For a Schwarz function 𝑤(𝑧) = 𝑐1𝑧+ 𝑐2𝑧2 + . . . , and for any 𝜇∈C we have ⃒⃒𝑐2 −𝜇𝑐2 1 ⃒⃒≤max {1, |𝜇|} . 3. Coefficient estimates for 𝑓∈𝑆*(q).
Theorem 1.
Theorem 1. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 +..., then |𝑎5| ≤13/24, |𝑎6| ≤29/30, |𝑎7| ≤309/288.
Theorem 1. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + . . . , then |𝑎5| ≤13/24, |𝑎6| ≤29/30, |𝑎7| ≤309/288.
Theorem 2.
Theorem 2. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+𝑎1𝑧+𝑎2𝑧2 +· · ·, then |𝑎2𝑎3 −𝑎4| ≤1. (38)
Theorem 2. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+𝑎1𝑧+𝑎2𝑧2 +· · ·, then |𝑎2𝑎3 −𝑎4| ≤1. (38)
Theorem 3.
Theorem 3. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+𝑎1𝑧+𝑎2𝑧2 +· · ·, then |𝐻3 (1) |≤265 192.
Theorem 3. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+𝑎1𝑧+𝑎2𝑧2 +· · ·, then |𝐻3 (1) |≤265 192.
Theorem 4.
Theorem 4. Let 𝑓−1 (𝑧) = 𝑧+ ∑︀∞ 𝑛=2 𝑑𝑛𝑧𝑛be the inverse function of 𝑓. For any 𝜇∈C and 𝑓∈𝑆* (q) of the form 𝑓(𝑧)=𝑧+𝑎1𝑧+𝑎2𝑧2+..., we get ⃒⃒𝑑3…
Theorem 4. Let 𝑓−1 (𝑧) = 𝑧+ ∑︀∞ 𝑛=2 𝑑𝑛𝑧𝑛be the inverse function of 𝑓. For any 𝜇∈C and 𝑓∈𝑆* (q) of the form 𝑓(𝑧)=𝑧+𝑎1𝑧+𝑎2𝑧2+. . . , we get ⃒⃒𝑑3 −𝜇𝑑2 2 ⃒⃒≤1 2 max {︂ 1, ⃒⃒⃒⃒ 5 −4𝜇 4 ⃒⃒⃒⃒ }︂ .
Theorem 5.
Theorem 5. For a function 𝑓∈𝑆* (q) of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ + 𝑎2𝑧2 +..., for any 𝜇∈C, and for 𝐺(𝑧) = 𝑧 𝑓(𝑧) = 1 + 𝑑1𝑧+ 𝑑2𝑧2 +..., we get…
Theorem 5. For a function 𝑓∈𝑆* (q) of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ + 𝑎2𝑧2 + . . . , for any 𝜇∈C, and for 𝐺(𝑧) = 𝑧 𝑓(𝑧) = 1 + 𝑑1𝑧+ 𝑑2𝑧2 + . . . , we get ⃒⃒𝑑2 −𝜇𝑑2 1 ⃒⃒≤1 2 max {︁ 1, ⃒⃒⃒1 −4𝜇 4 ⃒⃒⃒ }︁
Theorem 6.
Theorem 6. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 +..., then ⃒⃒𝑎3 −𝜇𝑎2 2 ⃒⃒≤ 1 2𝑔3 max ︁ 1, ⃒⃒⃒2𝜇𝑔3 −3𝑔2 2 2𝑔2 2 ⃒⃒⃒
Theorem 6. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + . . . , then ⃒⃒𝑎3 −𝜇𝑎2 2 ⃒⃒≤ 1 2𝑔3 max {︁ 1, ⃒⃒⃒2𝜇𝑔3 −3𝑔2 2 2𝑔2 2 ⃒⃒⃒
Theorem 7.
Theorem 7. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 +..., then ⃒⃒𝑎2 3 −𝑎5 ⃒⃒≤53 48.
Theorem 7. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + . . . , then ⃒⃒𝑎2 3 −𝑎5 ⃒⃒≤53 48.
Theorem 8.
Theorem 8. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 +..., then ⃒⃒𝑎2 4 −𝑎7 ⃒⃒≤127 96.
Theorem 8. If 𝑓∈𝑆* (q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + . . . , then ⃒⃒𝑎2 4 −𝑎7 ⃒⃒≤127 96 .
Theorem 9.
Theorem 9. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+𝑎1𝑧+𝑎2𝑧2 +· · ·, then |𝑎2| ≤1/2, |𝑎3| ≤1/4, |𝑎4| ≤7/24, |𝑎5| ≤3/40.
Theorem 9. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+𝑎1𝑧+𝑎2𝑧2 +· · ·, then |𝑎2| ≤1/2, |𝑎3| ≤1/4, |𝑎4| ≤7/24, |𝑎5| ≤3/40.
Theorem 10.
Theorem 10. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 +..., then |𝑎2𝑎3 −𝑎4| ≤29/48
Theorem 10. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + . . . , then |𝑎2𝑎3 −𝑎4| ≤29/48
Theorem 11.
Theorem 11. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 +..., then | 𝑎3 −𝜇𝑎2 2 |≤1 6 max ︁ 1, ⃒⃒⃒3(𝜇−1) 2 ⃒⃒⃒ ︁, (63) where 𝜇is a complex…
Theorem 11. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + . . . , then | 𝑎3 −𝜇𝑎2 2 |≤1 6 max {︁ 1, ⃒⃒⃒3(𝜇−1) 2 ⃒⃒⃒ }︁ , (63) where 𝜇is a complex number and the best bound is obtained.
Theorem 12.
Theorem 12. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 +..., then |𝑎2𝑎4 −𝑎2 3|≤31/144.
Theorem 12. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + . . . , then |𝑎2𝑎4 −𝑎2 3|≤31/144.
Theorem 13.
Theorem 13. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 +..., we have |𝐻3 (1) |≤1277 5760.
Theorem 13. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + . . . , we have |𝐻3 (1) |≤1277 5760.
Theorem 14.
Theorem 14. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + · · ·, then ⃒⃒𝑎2 3 −𝑎5 ⃒⃒≤7 48.
Theorem 14. If 𝑓∈𝒞(q) is of the form 𝑓(𝑧) = 𝑧+ 𝑎1𝑧+ 𝑎2𝑧2 + · · ·, then ⃒⃒𝑎2 3 −𝑎5 ⃒⃒≤7 48.
Definitions (2)
Def 1.
Definition 1. 𝑓∈𝒜is a function of the class 𝑆* (q) iff 𝑧𝑓′ (𝑧) 𝑓(𝑧) ≺ √ 1 + 𝑧2 + 𝑧. Definition 2. 𝑓, 𝑔∈𝒜are two functions of the class 𝑆*…
Definition 1. 𝑓∈𝒜is a function of the class 𝑆* (q) iff 𝑧𝑓′ (𝑧) 𝑓(𝑧) ≺ √ 1 + 𝑧2 + 𝑧. Definition 2. 𝑓, 𝑔∈𝒜are two functions of the class 𝑆* 𝑔(q) iff 𝑧 (︁
Def 3.
Definition 3. 𝑓∈𝒜is a function of the class 𝒞(q) iff (︁ 1 + 𝑧𝑓′′(𝑧) 𝑓′(𝑧) )︁ ≺ √ 1 + 𝑧2 + 𝑧. (5)
Definition 3. 𝑓∈𝒜is a function of the class 𝒞(q) iff (︁ 1 + 𝑧𝑓′′(𝑧) 𝑓′(𝑧) )︁ ≺ √ 1 + 𝑧2 + 𝑧. (5)
Function classes studied:
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