{"id":"47ca2791-4b35-4458-9aec-39153a022813","arxiv_id":"2607.12777","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Multidimensional analogues of improved Bohr inequalities are claimed for holomorphic functions on simply connected domains in C^n and for pluriharmonic mappings on polydisks containing the unit polydisk.","lead":"The paper claims sharper multidimensional Bohr inequalities for holomorphic functions on simply connected domains in several complex variables, improving a 2021 one-variable result, and a Bohr-type bound for pluriharmonic mappings on polydisks. Specialists in geometric function theory may use such radii to control power-series majorants in approximation and mapping problems.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the abstract-only limitation already noted by the reader.","rationale":"The reader's weakest_assumption correctly flags that the one-variable techniques may not extend without extra restrictions that shrink radii, and that this cannot be checked from the abstract. That remains the binding limitation. No stronger, more concrete technical concern (e.g., a specific equation or lemma that fails) can be extracted without the full text. Manufacturing a deeper attack would violate the good-faith and non-manufacturing rules. Therefore the UNVERDICTED / LOW-confidence posture is appropriate and should not be moved. The concrete_test is the minimal verification that would convert the information gap into either support or a genuine load-bearing objection.","tokens_in":1923,"tokens_out":443,"duration_ms":3976,"concrete_test":"Obtain the full preprint (or arXiv source) and verify that the multi-variable statements reduce exactly to the Evdoridis et al. (2021) one-variable improved Bohr radii when n=1 and the domain is a shifted disk, with no extra hypotheses that force a strictly smaller radius; if the reduction holds and the pluriharmonic polydisk radii are stated with explicit comparison to the unit-polydisk case, the abstract claims are supported at the level of statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper is available only as an abstract. The central claim is an improvement of Evdoridis et al. (2021) for a broader holomorphic class on simply connected domains in C^n, plus a Bohr-type inequality for pluriharmonic mappings on polydisks containing the unit polydisk. Without the body, precise function-class definitions, radii, lemmas, and comparisons cannot be checked for hidden restrictive hypotheses that would shrink the stated radii or for gaps in the multi-variable extension of the one-variable improved Bohr techniques. That is an information gap, not an identified internal inconsistency or load-bearing technical flaw. No concrete soft spot in the argument can be isolated from the abstract alone.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript investigates the Bohr phenomenon for holomorphic functions on general simply connected domains in C^n. It claims to improve the results of Evdoridis et al. (Results Math. 76, 14, 2021) for a broader class of holomorphic functions in several complex variables, and further establishes a Bohr-type inequality for pluriharmonic mappings defined on a polydisk containing the unit polydisk PΔ(0_n, 1_n).","tokens_in":2028,"tokens_out":677,"duration_ms":14015,"significance":"If the claimed improvements hold with the stated generality and without hidden restrictions that shrink the radii, the work would extend the improved Bohr inequality of Evdoridis et al. from the one-variable shifted-disk setting to a multi-variable holomorphic class on simply connected domains in C^n and to pluriharmonic mappings on enlarged polydisks. Such extensions are of genuine interest in geometric function theory. The abstract alone does not allow confirmation of sharpness, explicit radii, or the breadth of the function class, so significance remains conditional on the body of the paper.","major_comments":[{"comment":"Only the abstract is available for review. The central claims—an improvement of Evdoridis et al. (2021) for a broader holomorphic class on simply connected domains in C^n, and a Bohr-type inequality for pluriharmonic mappings on polydisks containing the unit polydisk—cannot be checked for correctness of proofs, precision of the radii obtained, or the presence of restrictive hypotheses that would limit the stated generality. A full assessment requires the complete manuscript with theorems, lemmas, and comparisons.","section":"Manuscript body (unavailable)"},{"comment":"The abstract asserts an extension of one-variable improved Bohr techniques to a broader multi-variable class and to pluriharmonic mappings without indicating whether additional hypotheses are imposed. Without the precise function-class definitions and the statements of the main theorems, it is impossible to verify that the multi-variable extension does not shrink the radii relative to the one-variable results being improved.","section":"Abstract (central claims)"}],"minor_comments":[{"comment":"Typographical errors in the abstract: 'Bhor phenomenon' should read 'Bohr phenomenon'; 'Evdordis et al.' should read 'Evdoridis et al.' (consistent with the cited Results Math. 76, 14 (2021)).","section":"Abstract"},{"comment":"Notation for the unit polydisk is written PΔ(0_n, 1_n); a brief clarification of this symbol (and consistency with standard polydisk notation) would aid readability once the full text is available.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review. The arXiv identifier 2607.12777 appears future-dated or placeholder; without the full text a standard referee report cannot be completed. I recommend requesting the complete manuscript before any editorial decision. No internal inconsistency is visible from the abstract alone; the limitation is purely informational."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is an abstract-only look at a pure complex-analysis preprint on multidimensional Bohr inequalities. The one thing to know: it claims an improvement of Evdoridis et al. (Results Math. 76, 14, 2021) for a broader class of holomorphic functions on general simply connected domains in C^n, plus a Bohr-type inequality for pluriharmonic mappings on a polydisk containing the unit polydisk. That is the whole contribution as stated.\n\nWhat is new, if true, is the multi-variable extension and the pluriharmonic piece. The program itself is established; this is a legitimate next step inside geometric function theory / several complex variables, not a new framework. Circularity looks low from the abstract—analytic majorant estimates building on prior work, no free parameters or invented entities visible. Pure-math inequality papers of this type are often sound when fully written, so I am not manufacturing a flaw.\n\nThe soft spot is purely informational. We cannot see the precise function-class definitions, the radii, the lemmas, or the comparison with the broader literature. The stress-test is right: the risk is that the one-variable improved Bohr techniques do not extend without extra restrictive hypotheses that shrink the stated radii. That is an information gap, not an identified internal inconsistency. Significance is limited-scope progress inside a specialized subfield; it does not reorganize anything.\n\nWho it is for: people already working on Bohr radii, pluriharmonic mappings, and multi-variable geometric function theory. A general complex analyst will not get much from the abstract alone. I would not bring it to reading group until the body is available, and I would not cite it yet. It still deserves a serious referee if the full text appears with clean statements and proofs—desk-rejecting abstract-only extensions of this kind is premature. Send it to peer review when the manuscript is complete; expect the referee to check the multi-variable hypotheses carefully.","headline":"Abstract-only multidimensional Bohr extension of Evdoridis et al.; honest subfield progress if the body holds, but nothing checkable yet.","tokens_in":2694,"tokens_out":492,"would_cite":false,"duration_ms":4608,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32A05","30C80","32A10"],"pacs":[],"model":"grok-4.5","headline":"Improved Bohr radii for holomorphic functions on general simply connected domains in C^n and for pluriharmonic mappings on enlarged polydisks.","keywords":["Bohr inequality","multidimensional Bohr radius","holomorphic functions","pluriharmonic mappings","polydisk","simply connected domains","several complex variables"],"falsifier":"An explicit holomorphic function on a simply connected domain in C^n (or a pluriharmonic mapping on a polydisk containing the unit polydisk) whose coefficient majorant exceeds the function's supremum inside every polydisk larger than the radius claimed by the paper.","tokens_in":2791,"feed_emoji":"📐","tokens_out":742,"duration_ms":6933,"temperature":0.7,"pith_summary":"The paper extends classical Bohr-type majorant inequalities from one complex variable into several variables. It claims that holomorphic functions on general simply connected domains in C^n obey improved Bohr radii of the same flavour that Evdoridis and co-authors obtained for shifted disks, but now for a strictly larger class of functions. Separately, it asserts that pluriharmonic mappings defined on a polydisk that properly contains the unit polydisk still satisfy a Bohr-type inequality. If the claims hold, they give concrete radius bounds that control the sum of the moduli of the coefficients by the supremum of the function, thereby enlarging the geometric settings in which such coefficient majorants are known to be valid.","feed_headline":"Bohr radii improved for holomorphic maps on domains in C^n","feed_subtitle":"Broader function class and pluriharmonic mappings on enlarged polydisks get explicit majorant bounds","key_machinery":"The multidimensional Bohr majorant inequality itself: an upper bound, valid inside a certain polydisk or domain of positive radius, that replaces the values of a holomorphic or pluriharmonic mapping by the sum of the moduli of its power-series coefficients, controlled by the supremum of the mapping.","core_discovery":"Holomorphic functions belonging to a broader class than previously treated, defined on general simply connected domains in C^n, obey improved multidimensional Bohr inequalities that refine the earlier results of Evdoridis et al. for shifted disks; in addition, pluriharmonic mappings on any polydisk containing the unit polydisk satisfy a Bohr-type majorant inequality.","pith_inferences":["The same majorant technique may adapt to other classes of mappings (e.g., harmonic or quasiregular) once corresponding one-variable Bohr radii are known.","If the radii turn out to be sharp, the paper would supply extremal examples that could serve as benchmarks for future multidimensional Bohr problems.","The results suggest a possible dictionary between geometric properties of the domain and the size of the Bohr radius in several complex variables."],"forward_implications":["Bohr radii previously known only for shifted disks become available for holomorphic functions on arbitrary simply connected domains in several complex variables.","The admissible class of holomorphic functions is strictly larger than the class treated by Evdoridis et al. (2021).","Pluriharmonic mappings on any polydisk containing the unit polydisk admit a Bohr-type majorant inequality.","Coefficient-sum estimates can be used on domains that properly contain the classical unit polydisk."],"fun_headline_variants":["Improved Bohr inequalities for broader holomorphic maps on C^n domains","Refined multidimensional Bohr bounds for shifted polydisks","Bohr-type majorants for pluriharmonic maps on enlarged polydisks","Broader holomorphic class yields sharper Bohr radii in C^n","Multidimensional Bohr refinements for domains and pluriharmonics"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The one-variable improved Bohr techniques of Evdoridis et al. extend to the claimed broader holomorphic class on simply connected domains in C^n and to pluriharmonic mappings on larger polydisks without extra hypotheses that would force the stated radii to shrink.","fun_headline_variants_meta":{"raw":{"variants":["Improved Bohr inequalities for broader holomorphic maps on C^n domains","Refined multidimensional Bohr bounds for shifted polydisks","Bohr-type majorants for pluriharmonic maps on enlarged polydisks","Broader holomorphic class yields sharper Bohr radii in C^n","Multidimensional Bohr refinements for domains and pluriharmonics"]},"model":"grok-4.5","effort":"low","cost_usd":0.002946,"raw_usage":{"total_tokens":985,"prompt_tokens":645,"num_sources_used":0,"completion_tokens":91,"cost_in_usd_ticks":29460000,"prompt_tokens_details":{"text_tokens":645,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":249,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":645,"tokens_out":91,"duration_ms":2419,"temperature":1.0,"reasoning_tokens":249,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-15T03:25:12.162419+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit holomorphic function on a simply connected domain in C^n (or a pluriharmonic mapping on a polydisk containing the unit polydisk) whose coefficient majorant exceeds the function's supremum inside every polydisk larger than the radius claimed by the paper.","supporting_citations":[],"review_version":1}